ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elvvuni GIF version

Theorem elvvuni 4708
Description: An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
Assertion
Ref Expression
elvvuni (𝐴 ∈ (V × V) → 𝐴𝐴)

Proof of Theorem elvvuni
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 4706 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 2755 . . . . . 6 𝑥 ∈ V
3 vex 2755 . . . . . 6 𝑦 ∈ V
42, 3uniop 4273 . . . . 5 𝑥, 𝑦⟩ = {𝑥, 𝑦}
52, 3opi2 4251 . . . . 5 {𝑥, 𝑦} ∈ ⟨𝑥, 𝑦
64, 5eqeltri 2262 . . . 4 𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦
7 unieq 3833 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
8 id 19 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = ⟨𝑥, 𝑦⟩)
97, 8eleq12d 2260 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → ( 𝐴𝐴𝑥, 𝑦⟩ ∈ ⟨𝑥, 𝑦⟩))
106, 9mpbiri 168 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴𝐴)
1110exlimivv 1908 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴𝐴)
121, 11sylbi 121 1 (𝐴 ∈ (V × V) → 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wex 1503  wcel 2160  Vcvv 2752  {cpr 3608  cop 3610   cuni 3824   × cxp 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-opab 4080  df-xp 4650
This theorem is referenced by:  unielxp  6200
  Copyright terms: Public domain W3C validator