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Theorem canth2g 7691
 Description: Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97. (Contributed by NM, 7-Nov-2003.)
Assertion
Ref Expression
canth2g

Proof of Theorem canth2g
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pweq 4015 . . 3
2 breq12 4457 . . 3
31, 2mpdan 668 . 2
4 vex 3112 . . 3
54canth2 7690 . 2
63, 5vtoclg 3167 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  e.wcel 1818  ~Pcpw 4012   class class class wbr 4452   csdm 7535 This theorem is referenced by:  2pwuninel  7692  2pwne  7693  pwfi  7835  cdalepw  8597  isfin32i  8766  fin34  8791  hsmexlem1  8827  canth3  8957  ondomon  8959  gchdomtri  9028  canthp1lem1  9051  canthp1lem2  9052  pwfseqlem5  9062  gchcdaidm  9067  gchxpidm  9068  gchpwdom  9069  gchaclem  9077  gchhar  9078  tsksdom  9155 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-en 7537  df-dom 7538  df-sdom 7539
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