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Theorem canth2g 7015
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 3628 . . 3  |-  ( x  =  A  ->  ~P x  =  ~P A
)
2 breq12 4028 . . 3  |-  ( ( x  =  A  /\  ~P x  =  ~P A )  ->  (
x  ~<  ~P x  <->  A  ~<  ~P A ) )
31, 2mpdan 649 . 2  |-  ( x  =  A  ->  (
x  ~<  ~P x  <->  A  ~<  ~P A ) )
4 vex 2791 . . 3  |-  x  e. 
_V
54canth2 7014 . 2  |-  x  ~<  ~P x
63, 5vtoclg 2843 1  |-  ( A  e.  V  ->  A  ~<  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   ~Pcpw 3625   class class class wbr 4023    ~< csdm 6862
This theorem is referenced by:  2pwuninel  7016  2pwne  7017  pwfi  7151  cdalepw  7822  isfin32i  7991  fin34  8016  hsmexlem1  8052  canth3  8183  ondomon  8185  gchdomtri  8251  canthp1lem1  8274  canthp1lem2  8275  pwfseqlem5  8285  gchcdaidm  8290  gchxpidm  8291  gchaclem  8292  gchhar  8293  gchpwdom  8296  tsksdom  8378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-en 6864  df-dom 6865  df-sdom 6866
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