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Theorem canth2g 7252
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  |-  ( A  e.  V  ->  A  ~<  ~P A )

Proof of Theorem canth2g
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pweq 3794 . . 3  |-  ( x  =  A  ->  ~P x  =  ~P A
)
2 breq12 4209 . . 3  |-  ( ( x  =  A  /\  ~P x  =  ~P A )  ->  (
x  ~<  ~P x  <->  A  ~<  ~P A ) )
31, 2mpdan 650 . 2  |-  ( x  =  A  ->  (
x  ~<  ~P x  <->  A  ~<  ~P A ) )
4 vex 2951 . . 3  |-  x  e. 
_V
54canth2 7251 . 2  |-  x  ~<  ~P x
63, 5vtoclg 3003 1  |-  ( A  e.  V  ->  A  ~<  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   ~Pcpw 3791   class class class wbr 4204    ~< csdm 7099
This theorem is referenced by:  2pwuninel  7253  2pwne  7254  pwfi  7393  cdalepw  8065  isfin32i  8234  fin34  8259  hsmexlem1  8295  canth3  8425  ondomon  8427  gchdomtri  8493  canthp1lem1  8516  canthp1lem2  8517  pwfseqlem5  8527  gchcdaidm  8532  gchxpidm  8533  gchaclem  8534  gchhar  8535  gchpwdom  8538  tsksdom  8620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-en 7101  df-dom 7102  df-sdom 7103
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