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Theorem canth2g 8733
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 4514 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2 breq12 5045 . . 3 ((𝑥 = 𝐴 ∧ 𝒫 𝑥 = 𝒫 𝐴) → (𝑥 ≺ 𝒫 𝑥𝐴 ≺ 𝒫 𝐴))
31, 2mpdan 687 . 2 (𝑥 = 𝐴 → (𝑥 ≺ 𝒫 𝑥𝐴 ≺ 𝒫 𝐴))
4 vex 3404 . . 3 𝑥 ∈ V
54canth2 8732 . 2 𝑥 ≺ 𝒫 𝑥
63, 5vtoclg 3473 1 (𝐴𝑉𝐴 ≺ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1542  wcel 2114  𝒫 cpw 4498   class class class wbr 5040  csdm 8566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-en 8568  df-dom 8569  df-sdom 8570
This theorem is referenced by:  2pwuninel  8734  2pwne  8735  pwfiOLD  8904  djulepw  9704  isfin32i  9877  fin34  9902  hsmexlem1  9938  canth3  10073  ondomon  10075  gchdomtri  10141  canthp1lem1  10164  canthp1lem2  10165  pwfseqlem5  10175  gchdjuidm  10180  gchxpidm  10181  gchpwdom  10182  gchaclem  10190  gchhar  10191  tsksdom  10268
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