Theorem elaltxp 35910

Description: Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.)

Assertion

Ref	Expression
elaltxp	⊢ (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫)

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

Proof of Theorem elaltxp

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3484	. 2 ⊢ (𝑋 ∈ (𝐴 ×× 𝐵) → 𝑋 ∈ V)
2		altopex 35895	. . . . 5 ⊢ ⟪𝑥, 𝑦⟫ ∈ V
3		eleq1 2821	. . . . 5 ⊢ (𝑋 = ⟪𝑥, 𝑦⟫ → (𝑋 ∈ V ↔ ⟪𝑥, 𝑦⟫ ∈ V))
4	2, 3	mpbiri 258	. . . 4 ⊢ (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
5	4	a1i 11	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V))
6	5	rexlimivv 3188	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
7		eqeq1 2738	. . . 4 ⊢ (𝑧 = 𝑋 → (𝑧 = ⟪𝑥, 𝑦⟫ ↔ 𝑋 = ⟪𝑥, 𝑦⟫))
8	7	2rexbidv 3209	. . 3 ⊢ (𝑧 = 𝑋 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
9		df-altxp 35894	. . 3 ⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
10	8, 9	elab2g 3663	. 2 ⊢ (𝑋 ∈ V → (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
11	1, 6, 10	pm5.21nii 378	1 ⊢ (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∃wrex 3059  Vcvv 3463  ⟪caltop 35891   ×× caltxp 35892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rex 3060  df-v 3465  df-dif 3934  df-un 3936  df-nul 4314  df-sn 4607  df-pr 4609  df-altop 35893  df-altxp 35894

This theorem is referenced by:  altopelaltxp  35911  altxpsspw  35912