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Theorem elvvuni 5065
 Description: An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
Assertion
Ref Expression
elvvuni

Proof of Theorem elvvuni
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 5063 . 2
2 vex 3112 . . . . . 6
3 vex 3112 . . . . . 6
42, 3uniop 4755 . . . . 5
52, 3opi2 4720 . . . . 5
64, 5eqeltri 2541 . . . 4
7 unieq 4257 . . . . 5
8 id 22 . . . . 5
97, 8eleq12d 2539 . . . 4
106, 9mpbiri 233 . . 3
1110exlimivv 1723 . 2
121, 11sylbi 195 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  =wceq 1395  E.wex 1612  e.wcel 1818   cvv 3109  {cpr 4031  <.cop 4035  U.cuni 4249  X.cxp 5002 This theorem is referenced by:  unielxp  6836 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-rex 2813  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-uni 4250  df-opab 4511  df-xp 5010
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