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Theorem canth2g 6969
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
StepHypRef Expression
1 pweq 3588 . . 3  |-  ( x  =  A  ->  ~P x  =  ~P A
)
2 breq12 3988 . . 3  |-  ( ( x  =  A  /\  ~P x  =  ~P A )  ->  (
x  ~<  ~P x  <->  A  ~<  ~P A ) )
31, 2mpdan 652 . 2  |-  ( x  =  A  ->  (
x  ~<  ~P x  <->  A  ~<  ~P A ) )
4 vex 2760 . . 3  |-  x  e. 
_V
54canth2 6968 . 2  |-  x  ~<  ~P x
63, 5vtoclg 2811 1  |-  ( A  e.  V  ->  A  ~<  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1619    e. wcel 1621   ~Pcpw 3585   class class class wbr 3983    ~< csdm 6816
This theorem is referenced by:  2pwuninel  6970  2pwne  6971  pwfi  7105  cdalepw  7776  isfin32i  7945  fin34  7970  hsmexlem1  8006  canth3  8137  ondomon  8139  gchdomtri  8205  canthp1lem1  8228  canthp1lem2  8229  pwfseqlem5  8239  gchcdaidm  8244  gchxpidm  8245  gchaclem  8246  gchhar  8247  gchpwdom  8250  tsksdom  8332
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-en 6818  df-dom 6819  df-sdom 6820
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