Theorem opkelxpkg 4248

Description: Kuratowski ordered pair membership in a Kuratowski cross product. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
opkelxpkg	⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ (C ×k D) ↔ (A ∈ C ∧ B ∈ D)))

Proof of Theorem opkelxpkg

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-xpk 4186	. 2 ⊢ (C ×k D) = {z ∣ ∃x∃y(z = ⟪x, y⟫ ∧ (x ∈ C ∧ y ∈ D))}
2		eleq1 2413	. . 3 ⊢ (x = A → (x ∈ C ↔ A ∈ C))
3	2	anbi1d 685	. 2 ⊢ (x = A → ((x ∈ C ∧ y ∈ D) ↔ (A ∈ C ∧ y ∈ D)))
4		eleq1 2413	. . 3 ⊢ (y = B → (y ∈ D ↔ B ∈ D))
5	4	anbi2d 684	. 2 ⊢ (y = B → ((A ∈ C ∧ y ∈ D) ↔ (A ∈ C ∧ B ∈ D)))
6	1, 3, 5	opkelopkabg 4246	1 ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ (C ×k D) ↔ (A ∈ C ∧ B ∈ D)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ⟪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-3an 936  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-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:  opkelxpk  4249  sikss1c1c  4268  opkelimagekg  4272