Theorem elxpi

Description: Membership in a Cartesian product. Uses fewer axioms than elxp . (Contributed by NM, 4-Jul-1994)

	Ref	Expression
Assertion	elxpi	⊢ ( 𝐴 ∈ ( 𝐵 × 𝐶 ) → ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq1	⊢ ( 𝑧 = 𝑤 → ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ↔ 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ) )
2	1	anbi1d	⊢ ( 𝑧 = 𝑤 → ( ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) ↔ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) ) )
3	2	2exbidv	⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) ) )
4		eqeq1	⊢ ( 𝑤 = 𝐴 → ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ↔ 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ) )
5	4	anbi1d	⊢ ( 𝑤 = 𝐴 → ( ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) ↔ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) ) )
6	5	2exbidv	⊢ ( 𝑤 = 𝐴 → ( ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) ) )
7		df-xp	⊢ ( 𝐵 × 𝐶 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) }
8		df-opab	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) } = { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) }
9	7 8	eqtri	⊢ ( 𝐵 × 𝐶 ) = { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) }
10	3 6 9	elab2gw	⊢ ( 𝐴 ∈ ( 𝐵 × 𝐶 ) → ( 𝐴 ∈ ( 𝐵 × 𝐶 ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) ) )
11	10	ibi	⊢ ( 𝐴 ∈ ( 𝐵 × 𝐶 ) → ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) )