Theorem elxp2 2443

Description: Membership in a cross product.

Assertion

Ref	Expression
elxp2	⊢ (A ∈ (B × C) ↔ ∃x ∈ B ∃y ∈ C A = ⟨x, y⟩)

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

Proof of Theorem elxp2

Step	Hyp	Ref	Expression
1		df-rex 1206	. . . 4 ⊢ (∃y ∈ C (x ∈ B ∧ A = ⟨x, y⟩) ↔ ∃y(y ∈ C ∧ (x ∈ B ∧ A = ⟨x, y⟩)))
2		r19.42v 1303	. . . 4 ⊢ (∃y ∈ C (x ∈ B ∧ A = ⟨x, y⟩) ↔ (x ∈ B ∧ ∃y ∈ C A = ⟨x, y⟩))
3		anass 336	. . . . . 6 ⊢ (((x ∈ B ∧ y ∈ C) ∧ A = ⟨x, y⟩) ↔ (x ∈ B ∧ (y ∈ C ∧ A = ⟨x, y⟩)))
4		ancom 333	. . . . . 6 ⊢ ((A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ ((x ∈ B ∧ y ∈ C) ∧ A = ⟨x, y⟩))
5		an12 370	. . . . . 6 ⊢ ((y ∈ C ∧ (x ∈ B ∧ A = ⟨x, y⟩)) ↔ (x ∈ B ∧ (y ∈ C ∧ A = ⟨x, y⟩)))
6	3, 4, 5	3bitr4r 159	. . . . 5 ⊢ ((y ∈ C ∧ (x ∈ B ∧ A = ⟨x, y⟩)) ↔ (A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
7	6	biex 733	. . . 4 ⊢ (∃y(y ∈ C ∧ (x ∈ B ∧ A = ⟨x, y⟩)) ↔ ∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
8	1, 2, 7	3bitr3 156	. . 3 ⊢ ((x ∈ B ∧ ∃y ∈ C A = ⟨x, y⟩) ↔ ∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
9	8	biex 733	. 2 ⊢ (∃x(x ∈ B ∧ ∃y ∈ C A = ⟨x, y⟩) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
10		df-rex 1206	. 2 ⊢ (∃x ∈ B ∃y ∈ C A = ⟨x, y⟩ ↔ ∃x(x ∈ B ∧ ∃y ∈ C A = ⟨x, y⟩))
11		elxp 2442	. 2 ⊢ (A ∈ (B × C) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
12	9, 10, 11	3bitr4r 159	1 ⊢ (A ∈ (B × C) ↔ ∃x ∈ B ∃y ∈ C A = ⟨x, y⟩)

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ⟨cop 1810   × cxp 2408

This theorem is referenced by:  xpdom2 3345  xpnnen 4927

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-rex 1206  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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