Theorem elaltxp

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

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

Proof

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