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Theorem elvvuni 4347
 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 4345 . 2 (A (V × V) ↔ xy A = ⟨x, y⟩)
2 vex 2554 . . . . . 6 x V
3 vex 2554 . . . . . 6 y V
42, 3uniop 3983 . . . . 5 x, y⟩ = {x, y}
52, 3opi2 3961 . . . . 5 {x, y} x, y
64, 5eqeltri 2107 . . . 4 x, yx, y
7 unieq 3580 . . . . 5 (A = ⟨x, y⟩ → A = x, y⟩)
8 id 19 . . . . 5 (A = ⟨x, y⟩ → A = ⟨x, y⟩)
97, 8eleq12d 2105 . . . 4 (A = ⟨x, y⟩ → ( A Ax, yx, y⟩))
106, 9mpbiri 157 . . 3 (A = ⟨x, y⟩ → A A)
1110exlimivv 1773 . 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 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  {cpr 3368  ⟨cop 3370  ∪ cuni 3571   × cxp 4286 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-opab 3810  df-xp 4294 This theorem is referenced by:  unielxp  5742
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