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Theorem canth2g 8918
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 4549 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2 breq12 5079 . . 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 3436 . . 3 𝑥 ∈ V
54canth2 8917 . 2 𝑥 ≺ 𝒫 𝑥
63, 5vtoclg 3505 1 (𝐴𝑉𝐴 ≺ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2106  𝒫 cpw 4533   class class class wbr 5074  csdm 8732
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-en 8734  df-dom 8735  df-sdom 8736
This theorem is referenced by:  2pwuninel  8919  2pwne  8920  pwfiOLD  9114  djulepw  9948  isfin32i  10121  fin34  10146  hsmexlem1  10182  canth3  10317  ondomon  10319  gchdomtri  10385  canthp1lem1  10408  canthp1lem2  10409  pwfseqlem5  10419  gchdjuidm  10424  gchxpidm  10425  gchpwdom  10426  gchaclem  10434  gchhar  10435  tsksdom  10512
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