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Theorem canthwdom 8026
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7690, equivalent to canth 6254). (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
canthwdom

Proof of Theorem canthwdom
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0elpw 4621 . . . . 5
2 ne0i 3790 . . . . 5
31, 2mp1i 12 . . . 4
4 brwdomn0 8016 . . . 4
53, 4syl 16 . . 3
65ibi 241 . 2
7 relwdom 8013 . . . . 5
87brrelex2i 5046 . . . 4
9 foeq2 5797 . . . . . . 7
10 pweq 4015 . . . . . . . 8
11 foeq3 5798 . . . . . . . 8
1210, 11syl 16 . . . . . . 7
139, 12bitrd 253 . . . . . 6
1413notbid 294 . . . . 5
15 vex 3112 . . . . . 6
1615canth 6254 . . . . 5
1714, 16vtoclg 3167 . . . 4
188, 17syl 16 . . 3
1918nexdv 1884 . 2
206, 19pm2.65i 173 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  <->wb 184  =wceq 1395  E.wex 1612  e.wcel 1818  =/=wne 2652   cvv 3109   c0 3784  ~Pcpw 4012   class class class wbr 4452  -onto->wfo 5591   cwdom 8004
This theorem is referenced by:  pwcdadom  8617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fo 5599  df-fv 5601  df-wdom 8006
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