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Theorem canthwdom 7547
 Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7260, equivalent to canth 6539). (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 4369 . . . . 5
2 ne0i 3634 . . . . 5
31, 2mp1i 12 . . . 4 *
4 brwdomn0 7537 . . . 4 *
53, 4syl 16 . . 3 * *
65ibi 233 . 2 *
7 relwdom 7534 . . . . 5 *
87brrelex2i 4919 . . . 4 *
9 foeq2 5650 . . . . . . 7
10 pweq 3802 . . . . . . . 8
11 foeq3 5651 . . . . . . . 8
1210, 11syl 16 . . . . . . 7
139, 12bitrd 245 . . . . . 6
1413notbid 286 . . . . 5
15 vex 2959 . . . . . 6
1615canth 6539 . . . . 5
1714, 16vtoclg 3011 . . . 4
188, 17syl 16 . . 3 *
1918nexdv 1941 . 2 *
206, 19pm2.65i 167 1 *
 Colors of variables: wff set class Syntax hints:   wn 3   wb 177  wex 1550   wceq 1652   wcel 1725   wne 2599  cvv 2956  c0 3628  cpw 3799   class class class wbr 4212  wfo 5452   * cwdom 7525 This theorem is referenced by:  pwcdadom  8096 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-wdom 7527
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