Theorem elxp3 2460

Description: Membership in a cross product.

Assertion

Ref	Expression
elxp3	⊢ (A ∈ (B × C) ↔ ∃x∃y(⟨x, y⟩ = A ∧ ⟨x, y⟩ ∈ (B × C)))

Distinct variable group(s):   x,y,A   x,B,y   x,C,y

Proof of Theorem elxp3

Step	Hyp	Ref	Expression
1		elxp 2442	. 2 ⊢ (A ∈ (B × C) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
2		cleqcom 1103	. . . 4 ⊢ (⟨x, y⟩ = A ↔ A = ⟨x, y⟩)
3		visset 1350	. . . . 5 ⊢ y ∈ V
4	3	opelxp 2452	. . . 4 ⊢ (⟨x, y⟩ ∈ (B × C) ↔ (x ∈ B ∧ y ∈ C))
5	2, 4	anbi12i 369	. . 3 ⊢ ((⟨x, y⟩ = A ∧ ⟨x, y⟩ ∈ (B × C)) ↔ (A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
6	5	bi2ex 734	. 2 ⊢ (∃x∃y(⟨x, y⟩ = A ∧ ⟨x, y⟩ ∈ (B × C)) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
7	1, 6	bitr4 154	1 ⊢ (A ∈ (B × C) ↔ ∃x∃y(⟨x, y⟩ = A ∧ ⟨x, y⟩ ∈ (B × C)))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   × cxp 2408

This theorem is referenced by:  optocl 2469  cbvop 2473

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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