Theorem elxpk2 4198

Description: Membership in a cross product. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
elxpk2	⊢ (A ∈ (B ×k C) ↔ ∃x ∈ B ∃y ∈ C A = ⟪x, y⟫)

Distinct variable groups:   x,A,y   x,B,y   x,C,y

Proof of Theorem elxpk2

Step	Hyp	Ref	Expression
1		ancom 437	. . 3 ⊢ (((x ∈ B ∧ y ∈ C) ∧ A = ⟪x, y⟫) ↔ (A = ⟪x, y⟫ ∧ (x ∈ B ∧ y ∈ C)))
2	1	2exbii 1583	. 2 ⊢ (∃x∃y((x ∈ B ∧ y ∈ C) ∧ A = ⟪x, y⟫) ↔ ∃x∃y(A = ⟪x, y⟫ ∧ (x ∈ B ∧ y ∈ C)))
3		r2ex 2653	. 2 ⊢ (∃x ∈ B ∃y ∈ C A = ⟪x, y⟫ ↔ ∃x∃y((x ∈ B ∧ y ∈ C) ∧ A = ⟪x, y⟫))
4		elxpk 4197	. 2 ⊢ (A ∈ (B ×k C) ↔ ∃x∃y(A = ⟪x, y⟫ ∧ (x ∈ B ∧ y ∈ C)))
5	2, 3, 4	3bitr4ri 269	1 ⊢ (A ∈ (B ×k C) ↔ ∃x ∈ B ∃y ∈ C A = ⟪x, y⟫)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ⟪copk 4058   ×_k cxpk 4175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059  df-xpk 4186

This theorem is referenced by:  xpkeq1  4199  xpkeq2  4200