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Theorem canth2g 9127
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 4608 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2 breq12 5143 . . 3 ((𝑥 = 𝐴 ∧ 𝒫 𝑥 = 𝒫 𝐴) → (𝑥 ≺ 𝒫 𝑥𝐴 ≺ 𝒫 𝐴))
31, 2mpdan 684 . 2 (𝑥 = 𝐴 → (𝑥 ≺ 𝒫 𝑥𝐴 ≺ 𝒫 𝐴))
4 vex 3470 . . 3 𝑥 ∈ V
54canth2 9126 . 2 𝑥 ≺ 𝒫 𝑥
63, 5vtoclg 3535 1 (𝐴𝑉𝐴 ≺ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  𝒫 cpw 4594   class class class wbr 5138  csdm 8934
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-en 8936  df-dom 8937  df-sdom 8938
This theorem is referenced by:  2pwuninel  9128  2pwne  9129  pwfiOLD  9343  djulepw  10183  isfin32i  10356  fin34  10381  hsmexlem1  10417  canth3  10552  ondomon  10554  gchdomtri  10620  canthp1lem1  10643  canthp1lem2  10644  pwfseqlem5  10654  gchdjuidm  10659  gchxpidm  10660  gchpwdom  10661  gchaclem  10669  gchhar  10670  tsksdom  10747
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