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Theorem canth2g 8663
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 (𝐴𝑉𝐴 ≺ 𝒫 𝐴)

Proof of Theorem canth2g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pweq 4544 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2 breq12 5067 . . 3 ((𝑥 = 𝐴 ∧ 𝒫 𝑥 = 𝒫 𝐴) → (𝑥 ≺ 𝒫 𝑥𝐴 ≺ 𝒫 𝐴))
31, 2mpdan 683 . 2 (𝑥 = 𝐴 → (𝑥 ≺ 𝒫 𝑥𝐴 ≺ 𝒫 𝐴))
4 vex 3502 . . 3 𝑥 ∈ V
54canth2 8662 . 2 𝑥 ≺ 𝒫 𝑥
63, 5vtoclg 3572 1 (𝐴𝑉𝐴 ≺ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2107  𝒫 cpw 4541   class class class wbr 5062  csdm 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-en 8502  df-dom 8503  df-sdom 8504
This theorem is referenced by:  2pwuninel  8664  2pwne  8665  pwfi  8811  djulepw  9610  isfin32i  9779  fin34  9804  hsmexlem1  9840  canth3  9975  ondomon  9977  gchdomtri  10043  canthp1lem1  10066  canthp1lem2  10067  pwfseqlem5  10077  gchdjuidm  10082  gchxpidm  10083  gchpwdom  10084  gchaclem  10092  gchhar  10093  tsksdom  10170
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