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Theorem canth2g 7011
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 )
Dummy variable  x is distinct from all other variables.

Proof of Theorem canth2g
StepHypRef Expression
1 pweq 3630 . . 3  |-  ( x  =  A  ->  ~P x  =  ~P A
)
2 breq12 4030 . . 3  |-  ( ( x  =  A  /\  ~P x  =  ~P A )  ->  (
x  ~<  ~P x  <->  A  ~<  ~P A ) )
31, 2mpdan 651 . 2  |-  ( x  =  A  ->  (
x  ~<  ~P x  <->  A  ~<  ~P A ) )
4 vex 2793 . . 3  |-  x  e. 
_V
54canth2 7010 . 2  |-  x  ~<  ~P x
63, 5vtoclg 2845 1  |-  ( A  e.  V  ->  A  ~<  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1624    e. wcel 1685   ~Pcpw 3627   class class class wbr 4025    ~< csdm 6858
This theorem is referenced by:  2pwuninel  7012  2pwne  7013  pwfi  7147  cdalepw  7818  isfin32i  7987  fin34  8012  hsmexlem1  8048  canth3  8179  ondomon  8181  gchdomtri  8247  canthp1lem1  8270  canthp1lem2  8271  pwfseqlem5  8281  gchcdaidm  8286  gchxpidm  8287  gchaclem  8288  gchhar  8289  gchpwdom  8292  tsksdom  8374
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-en 6860  df-dom 6861  df-sdom 6862
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