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Theorem canth2g 7297
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 3831 . . 3  |-  ( x  =  A  ->  ~P x  =  ~P A
)
2 breq12 4248 . . 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 2968 . . 3  |-  x  e. 
_V
54canth2 7296 . 2  |-  x  ~<  ~P x
63, 5vtoclg 3020 1  |-  ( A  e.  V  ->  A  ~<  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1654    e. wcel 1728   ~Pcpw 3828   class class class wbr 4243    ~< csdm 7144
This theorem is referenced by:  2pwuninel  7298  2pwne  7299  pwfi  7438  cdalepw  8114  isfin32i  8283  fin34  8308  hsmexlem1  8344  canth3  8474  ondomon  8476  gchdomtri  8542  canthp1lem1  8565  canthp1lem2  8566  pwfseqlem5  8576  gchcdaidm  8581  gchxpidm  8582  gchpwdom  8583  gchaclem  8591  gchhar  8592  tsksdom  8669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-en 7146  df-dom 7147  df-sdom 7148
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