Theorem altopthg

Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012)

	Ref	Expression
Assertion	altopthg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		altopthsn	⊢ ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ ↔ ( { 𝐴 } = { 𝐶 } ∧ { 𝐵 } = { 𝐷 } ) )
2		sneqbg	⊢ ( 𝐴 ∈ 𝑉 → ( { 𝐴 } = { 𝐶 } ↔ 𝐴 = 𝐶 ) )
3		sneqbg	⊢ ( 𝐵 ∈ 𝑊 → ( { 𝐵 } = { 𝐷 } ↔ 𝐵 = 𝐷 ) )
4	2 3	bi2anan9	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( { 𝐴 } = { 𝐶 } ∧ { 𝐵 } = { 𝐷 } ) ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) )
5	1 4	bitrid	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) )