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Theorem canthwdom 8469
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 8098, equivalent to canth 6593). (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
canthwdom ¬ 𝒫 𝐴* 𝐴

Proof of Theorem canthwdom
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0elpw 4825 . . . . 5 ∅ ∈ 𝒫 𝐴
2 ne0i 3913 . . . . 5 (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅)
31, 2mp1i 13 . . . 4 (𝒫 𝐴* 𝐴 → 𝒫 𝐴 ≠ ∅)
4 brwdomn0 8459 . . . 4 (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴))
53, 4syl 17 . . 3 (𝒫 𝐴* 𝐴 → (𝒫 𝐴* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴))
65ibi 256 . 2 (𝒫 𝐴* 𝐴 → ∃𝑓 𝑓:𝐴onto→𝒫 𝐴)
7 relwdom 8456 . . . . 5 Rel ≼*
87brrelex2i 5149 . . . 4 (𝒫 𝐴* 𝐴𝐴 ∈ V)
9 foeq2 6099 . . . . . . 7 (𝑥 = 𝐴 → (𝑓:𝑥onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝑥))
10 pweq 4152 . . . . . . . 8 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
11 foeq3 6100 . . . . . . . 8 (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
1210, 11syl 17 . . . . . . 7 (𝑥 = 𝐴 → (𝑓:𝐴onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
139, 12bitrd 268 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝑥onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
1413notbid 308 . . . . 5 (𝑥 = 𝐴 → (¬ 𝑓:𝑥onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴onto→𝒫 𝐴))
15 vex 3198 . . . . . 6 𝑥 ∈ V
1615canth 6593 . . . . 5 ¬ 𝑓:𝑥onto→𝒫 𝑥
1714, 16vtoclg 3261 . . . 4 (𝐴 ∈ V → ¬ 𝑓:𝐴onto→𝒫 𝐴)
188, 17syl 17 . . 3 (𝒫 𝐴* 𝐴 → ¬ 𝑓:𝐴onto→𝒫 𝐴)
1918nexdv 1862 . 2 (𝒫 𝐴* 𝐴 → ¬ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴)
206, 19pm2.65i 185 1 ¬ 𝒫 𝐴* 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1481  wex 1702  wcel 1988  wne 2791  Vcvv 3195  c0 3907  𝒫 cpw 4149   class class class wbr 4644  ontowfo 5874  * cwdom 8447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884  df-wdom 8449
This theorem is referenced by:  pwcdadom  9023
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