Theorem elxpk 4197

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

Assertion

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

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

Proof of Theorem elxpk

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2868	. 2 ⊢ (A ∈ (B ×k C) → A ∈ V)
2		opkex 4114	. . . . 5 ⊢ ⟪x, y⟫ ∈ V
3		eleq1 2413	. . . . 5 ⊢ (A = ⟪x, y⟫ → (A ∈ V ↔ ⟪x, y⟫ ∈ V))
4	2, 3	mpbiri 224	. . . 4 ⊢ (A = ⟪x, y⟫ → A ∈ V)
5	4	adantr 451	. . 3 ⊢ ((A = ⟪x, y⟫ ∧ (x ∈ B ∧ y ∈ C)) → A ∈ V)
6	5	exlimivv 1635	. 2 ⊢ (∃x∃y(A = ⟪x, y⟫ ∧ (x ∈ B ∧ y ∈ C)) → A ∈ V)
7		eqeq1 2359	. . . . 5 ⊢ (w = A → (w = ⟪x, y⟫ ↔ A = ⟪x, y⟫))
8	7	anbi1d 685	. . . 4 ⊢ (w = A → ((w = ⟪x, y⟫ ∧ (x ∈ B ∧ y ∈ C)) ↔ (A = ⟪x, y⟫ ∧ (x ∈ B ∧ y ∈ C))))
9	8	2exbidv 1628	. . 3 ⊢ (w = A → (∃x∃y(w = ⟪x, y⟫ ∧ (x ∈ B ∧ y ∈ C)) ↔ ∃x∃y(A = ⟪x, y⟫ ∧ (x ∈ B ∧ y ∈ C))))
10		df-xpk 4186	. . 3 ⊢ (B ×k C) = {w ∣ ∃x∃y(w = ⟪x, y⟫ ∧ (x ∈ B ∧ y ∈ C))}
11	9, 10	elab2g 2988	. 2 ⊢ (A ∈ V → (A ∈ (B ×k C) ↔ ∃x∃y(A = ⟪x, y⟫ ∧ (x ∈ B ∧ y ∈ C))))
12	1, 6, 11	pm5.21nii 342	1 ⊢ (A ∈ (B ×k C) ↔ ∃x∃y(A = ⟪x, y⟫ ∧ (x ∈ B ∧ y ∈ C)))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟪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-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:  elxpk2  4198  elvvk  4208  xpkvexg  4286  sikexlem  4296  insklem  4305