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Theorem elvvuni 4930
 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 4928 . 2
2 vex 2951 . . . . . 6
3 vex 2951 . . . . . 6
42, 3uniop 4451 . . . . 5
52, 3opi2 4423 . . . . 5
64, 5eqeltri 2505 . . . 4
7 unieq 4016 . . . . 5
8 id 20 . . . . 5
97, 8eleq12d 2503 . . . 4
106, 9mpbiri 225 . . 3
1110exlimivv 1645 . 2
121, 11sylbi 188 1
 Colors of variables: wff set class Syntax hints:   wi 4  wex 1550   wceq 1652   wcel 1725  cvv 2948  cpr 3807  cop 3809  cuni 4007   cxp 4868 This theorem is referenced by:  unielxp  6377 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-opab 4259  df-xp 4876
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