Theorem opelxpex 2445

Description: The first member of an ordered pair of classes in a cross product exists. (This is a byproduct of our definition of ordered pair. Unfortunately existence is not implied for the second member.)

Assertion

Ref	Expression
opelxpex	⊢ (⟨A, B⟩ ∈ (C × D) → A ∈ V)

Proof of Theorem opelxpex

Step	Hyp	Ref	Expression
1		elxp 2442	. 2 ⊢ (⟨A, B⟩ ∈ (C × D) ↔ ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ (x ∈ C ∧ y ∈ D)))
2		visset 1350	. . . . 5 ⊢ x ∈ V
3		eleq2 1150	. . . . . . 7 ⊢ (⟨A, B⟩ = ⟨x, y⟩ → (∅ ∈ ⟨A, B⟩ ↔ ∅ ∈ ⟨x, y⟩))
4		opprc1b 1906	. . . . . . 7 ⊢ (¬ A ∈ V ↔ ∅ ∈ ⟨A, B⟩)
5		opprc1b 1906	. . . . . . 7 ⊢ (¬ x ∈ V ↔ ∅ ∈ ⟨x, y⟩)
6	3, 4, 5	3bitr4g 428	. . . . . 6 ⊢ (⟨A, B⟩ = ⟨x, y⟩ → (¬ A ∈ V ↔ ¬ x ∈ V))
7	6	bicon4d 402	. . . . 5 ⊢ (⟨A, B⟩ = ⟨x, y⟩ → (A ∈ V ↔ x ∈ V))
8	2, 7	mpbiri 169	. . . 4 ⊢ (⟨A, B⟩ = ⟨x, y⟩ → A ∈ V)
9	8	adantr 306	. . 3 ⊢ ((⟨A, B⟩ = ⟨x, y⟩ ∧ (x ∈ C ∧ y ∈ D)) → A ∈ V)
10	9	19.23aivv 953	. 2 ⊢ (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ (x ∈ C ∧ y ∈ D)) → A ∈ V)
11	1, 10	sylbi 174	1 ⊢ (⟨A, B⟩ ∈ (C × D) → A ∈ V)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  ⟨cop 1810   × cxp 2408

This theorem is referenced by:  brrelex 2446  opelxp 2452  imasn 2616

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424

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