Theorem elaltxp 36506

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 3478	. 2 ⊢ (𝑋 ∈ (𝐴 ×× 𝐵) → 𝑋 ∈ V)
2		altopex 36491	. . . . 5 ⊢ ⟪𝑥, 𝑦⟫ ∈ V
3		eleq1 2853	. . . . 5 ⊢ (𝑋 = ⟪𝑥, 𝑦⟫ → (𝑋 ∈ V ↔ ⟪𝑥, 𝑦⟫ ∈ V))
4	2, 3	mpbiri 261	. . . 4 ⊢ (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
5	4	a1i 11	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V))
6	5	rexlimivv 3209	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
7		eqeq1 2769	. . . 4 ⊢ (𝑧 = 𝑋 → (𝑧 = ⟪𝑥, 𝑦⟫ ↔ 𝑋 = ⟪𝑥, 𝑦⟫))
8	7	2rexbidv 3232	. . 3 ⊢ (𝑧 = 𝑋 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
9		df-altxp 36490	. . 3 ⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
10	8, 9	elab2g 3641	. 2 ⊢ (𝑋 ∈ V → (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
11	1, 6, 10	pm5.21nii 381	1 ⊢ (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑋 = ⟪𝑥, 𝑦⟫)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  Vcvv 3457  ⟪caltop 36487   ×× caltxp 36488

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rex 3092  df-v 3459  df-un 3911  df-sn 4592  df-pr 4594  df-altop 36489  df-altxp 36490

This theorem is used by:  altopelaltxp  36507  altxpsspw  36508