Theorem elaltxp 34277

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 3450	. 2 ⊢ (𝑋 ∈ (𝐴 ×× 𝐵) → 𝑋 ∈ V)
2		altopex 34262	. . . . 5 ⊢ ⟪𝑥, 𝑦⟫ ∈ V
3		eleq1 2826	. . . . 5 ⊢ (𝑋 = ⟪𝑥, 𝑦⟫ → (𝑋 ∈ V ↔ ⟪𝑥, 𝑦⟫ ∈ V))
4	2, 3	mpbiri 257	. . . 4 ⊢ (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
5	4	a1i 11	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V))
6	5	rexlimivv 3221	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
7		eqeq1 2742	. . . 4 ⊢ (𝑧 = 𝑋 → (𝑧 = ⟪𝑥, 𝑦⟫ ↔ 𝑋 = ⟪𝑥, 𝑦⟫))
8	7	2rexbidv 3229	. . 3 ⊢ (𝑧 = 𝑋 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
9		df-altxp 34261	. . 3 ⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
10	8, 9	elab2g 3611	. 2 ⊢ (𝑋 ∈ V → (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
11	1, 6, 10	pm5.21nii 380	1 ⊢ (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432  ⟪caltop 34258   ×× caltxp 34259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564  df-altop 34260  df-altxp 34261

This theorem is referenced by:  altopelaltxp  34278  altxpsspw  34279