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Theorem elvvuni 4331
Description: An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
Assertion
Ref Expression
elvvuni (A (V × V) → A A)

Proof of Theorem elvvuni
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 4329 . 2 (A (V × V) ↔ xy A = ⟨x, y⟩)
2 vex 2538 . . . . . 6 x V
3 vex 2538 . . . . . 6 y V
42, 3uniop 3966 . . . . 5 x, y⟩ = {x, y}
52, 3opi2 3944 . . . . 5 {x, y} x, y
64, 5eqeltri 2092 . . . 4 x, yx, y
7 unieq 3563 . . . . 5 (A = ⟨x, y⟩ → A = x, y⟩)
8 id 19 . . . . 5 (A = ⟨x, y⟩ → A = ⟨x, y⟩)
97, 8eleq12d 2090 . . . 4 (A = ⟨x, y⟩ → ( A Ax, yx, y⟩))
106, 9mpbiri 157 . . 3 (A = ⟨x, y⟩ → A A)
1110exlimivv 1758 . 2 (xy A = ⟨x, y⟩ → A A)
121, 11sylbi 114 1 (A (V × V) → A A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228  wex 1362   wcel 1374  Vcvv 2535  {cpr 3351  cop 3353   cuni 3554   × cxp 4270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-opab 3793  df-xp 4278
This theorem is referenced by:  unielxp  5723
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