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Theorem elvvv 5064
Description: Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
Assertion
Ref Expression
elvvv
Distinct variable group:   , , ,

Proof of Theorem elvvv
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elxp 5021 . 2
2 anass 649 . . . . 5
3 19.42vv 1777 . . . . . 6
4 ancom 450 . . . . . . 7
542exbii 1668 . . . . . 6
6 vex 3112 . . . . . . . 8
76biantru 505 . . . . . . 7
8 elvv 5063 . . . . . . . 8
98anbi2i 694 . . . . . . 7
107, 9bitr3i 251 . . . . . 6
113, 5, 103bitr4ri 278 . . . . 5
122, 11bitr3i 251 . . . 4
13122exbii 1668 . . 3
14 exrot4 1853 . . . 4
15 excom 1849 . . . . . 6
16 opex 4716 . . . . . . . 8
17 opeq1 4217 . . . . . . . . 9
1817eqeq2d 2471 . . . . . . . 8
1916, 18ceqsexv 3146 . . . . . . 7
2019exbii 1667 . . . . . 6
2115, 20bitri 249 . . . . 5
22212exbii 1668 . . . 4
2314, 22bitr3i 251 . . 3
2413, 23bitri 249 . 2
251, 24bitri 249 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818   cvv 3109  <.cop 4035  X.cxp 5002
This theorem is referenced by:  ssrelrel  5108  dftpos3  6992
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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  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-opab 4511  df-xp 5010
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