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Theorem canth2g 9078
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 4575 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2 breq12 5111 . . 3 ((𝑥 = 𝐴 ∧ 𝒫 𝑥 = 𝒫 𝐴) → (𝑥 ≺ 𝒫 𝑥𝐴 ≺ 𝒫 𝐴))
31, 2mpdan 686 . 2 (𝑥 = 𝐴 → (𝑥 ≺ 𝒫 𝑥𝐴 ≺ 𝒫 𝐴))
4 vex 3448 . . 3 𝑥 ∈ V
54canth2 9077 . 2 𝑥 ≺ 𝒫 𝑥
63, 5vtoclg 3524 1 (𝐴𝑉𝐴 ≺ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  𝒫 cpw 4561   class class class wbr 5106  csdm 8885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-en 8887  df-dom 8888  df-sdom 8889
This theorem is referenced by:  2pwuninel  9079  2pwne  9080  pwfiOLD  9294  djulepw  10133  isfin32i  10306  fin34  10331  hsmexlem1  10367  canth3  10502  ondomon  10504  gchdomtri  10570  canthp1lem1  10593  canthp1lem2  10594  pwfseqlem5  10604  gchdjuidm  10609  gchxpidm  10610  gchpwdom  10611  gchaclem  10619  gchhar  10620  tsksdom  10697
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