diff --git a/set.mm b/set.mm index bd3025843..e938d7c45 100644 --- a/set.mm +++ b/set.mm @@ -33834,6 +33834,18 @@ general is seen either by substitution (when the variable ` V ` has no ( wcel cv wceq wa wrex r19.42v eleq1 adantr pm5.32ri bicomi baib ceqsrexv wb rexbiia pm5.32i 3bitr3i ) DEGZCHZDIZAJZJZCEKUCUFCEKZJUHUCBJUCUFCELUGUF CEUGUDEGZUFUIUFJUGUFUIUCUEUIUCSAUDDEMNOPQTUCUHBABCDEFRUAUB $. + + $( Alternate elimitation of a restricted existential quantifier, using + implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.) $) + ceqsrexv2 $p |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) ) $= + ( ceqsrexbv ) ABCDEFG $. + + $( Alternate elimination of a restricted universal quantifier, using + implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.) $) + ceqsralv2 $p |- ( A. x e. B ( x = A -> ph ) <-> ( A e. B -> ps ) ) $= + ( cv wceq wi wral wcel wn wa wrex notbid ceqsrexv2 rexanali annim 3bitr3i + con4bii ) CGDHZAICEJZDEKZBIZUAALZMCENUCBLZMUBLUDLUEUFCDEUAABFOPUAACEQUCBR + ST $. $} ${ @@ -532424,15 +532436,6 @@ Real and complex numbers (cont.) brtpid3 $p |- A { C , D , <. A , B >. } B $= ( cop ctp wbr wcel opex tpid3 df-br mpbir ) ABCDABEZFZGMNHCDMABIJABNKL $. - ${ - $d A x $. $d B x $. $d ps x $. - ceqsrexv2.1 $e |- ( x = A -> ( ph <-> ps ) ) $. - $( Alternate elimitation of a restricted existential quantifier, using - implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.) $) - ceqsrexv2 $p |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) ) $= - ( ceqsrexbv ) ABCDEFG $. - $} - ${ $d x V $. $d A y $. $d ps y $. $d x y $. iota5f.1 $e |- F/ x ph $. @@ -532447,17 +532450,6 @@ Real and complex numbers (cont.) GUEVEUTUNBVCDUMTUBUFVCDUQTUCBCIUGUHUIUJ $. $} - ${ - $d x A $. $d x B $. $d ps x $. - ceqsralv2.1 $e |- ( x = A -> ( ph <-> ps ) ) $. - $( Alternate elimination of a restricted universal quantifier, using - implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.) $) - ceqsralv2 $p |- ( A. x e. B ( x = A -> ph ) <-> ( A e. B -> ps ) ) $= - ( cv wceq wi wral wcel wn wa wrex notbid ceqsrexv2 rexanali annim 3bitr3i - con4bii ) CGDHZAICEJZDEKZBIZUAALZMCENUCBLZMUBLUDLUEUFCDEUAABFOPUAACEQUCBR - ST $. - $} - $( A class is ordinal iff it is a subclass of ` On ` and transitive. (Contributed by Scott Fenton, 21-Nov-2021.) $) dford5 $p |- ( Ord A <-> ( A C_ On /\ Tr A ) ) $= @@ -582175,38 +582167,16 @@ equivalence of domain elementhood (equivalence is not necessary as HUFUEUHPBUCUDQRSTUBUA $. $} - ${ - $d A x $. $d B x $. $d ps x $. - ceqsrexv2TEMP.1 $e |- ( x = A -> ( ph <-> ps ) ) $. - $( Alternate elimitation of a restricted existential quantifier, using - implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.) $) - ceqsrexv2TEMP $p |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) ) $= - ( ceqsrexbv ) ABCDEFG $. - $} - - ${ - $d x A $. $d x B $. $d ps x $. - ceqsralv2TEMP.1 $e |- ( x = A -> ( ph <-> ps ) ) $. - $( Alternate elimination of a restricted universal quantifier, using - implicit substitution. (Temporary: as soon as this Mathbox only PR is - accepted, I'll open a PR to place this to the main. PM) (Contributed by - Scott Fenton, 7-Dec-2020.) $) - ceqsralv2TEMP $p |- ( A. x e. B ( x = A -> ph ) <-> ( A e. B -> ps ) ) $= - ( cv wceq wi wral wcel wrex notbid ceqsrexv2TEMP rexanali 3bitr3i con4bii - wn wa annim ) CGDHZAICEJZDEKZBIZUAARZSCELUCBRZSUBRUDRUEUFCDEUAABFMNUAACEO - UCBTPQ $. - $} - ${ $d A x y $. $d B x y $. $d R x y $. $( Two ways to say that an intersection of the identity relation with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 12-Dec-2023.) $) ref5 $p |- ( ( _I i^i ( A X. B ) ) C_ R <-> A. x e. ( A i^i B ) x R x ) $= - ( vy cv wceq wbr wi wral wcel cid cxp cin wss equcom imbi1i ceqsralv2TEMP - ralbii breq2 bitr3i idinxpss ralin 3bitr4i ) AFZEFZGZUEUFDHZIZECJZABJUECK - UEUEDHZIZABJLBCMNDOUKABCNJUJULABUJUFUEGZUHIZECJULUNUIECUMUGUHEAPQSUHUKEUE - CUFUEUEDTRUASAEBCDUBUKABCUCUD $. + ( vy cv wceq wbr wral wcel cid cxp cin wss equcom imbi1i ralbii ceqsralv2 + wi breq2 bitr3i idinxpss ralin 3bitr4i ) AFZEFZGZUEUFDHZSZECIZABIUECJUEUE + DHZSZABIKBCLMDNUKABCMIUJULABUJUFUEGZUHSZECIULUNUIECUMUGUHEAOPQUHUKEUECUFU + EUEDTRUAQAEBCDUBUKABCUCUD $. $} ${