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Theorem ecdmn0 7373
 Description: A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecdmn0

Proof of Theorem ecdmn0
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3118 . 2
2 n0 3794 . . 3
3 ecexr 7335 . . . 4
43exlimiv 1722 . . 3
52, 4sylbi 195 . 2
6 vex 3112 . . . . 5
7 elecg 7369 . . . . 5
86, 7mpan 670 . . . 4
98exbidv 1714 . . 3
102a1i 11 . . 3
11 eldmg 5203 . . 3
129, 10, 113bitr4rd 286 . 2
131, 5, 12pm5.21nii 353 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  E.wex 1612  e.wcel 1818  =/=wne 2652   cvv 3109   c0 3784   class class class wbr 4452  domcdm 5004  [cec 7328 This theorem is referenced by:  ereldm  7374  elqsn0  7399  ecelqsdm  7400  eceqoveq  7435  divsfval  14944  sylow1lem5  16622  vitalilem2  22018  vitalilem3  22019 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-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 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-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-ec 7332
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