Theorem bj-2uplth

Description: The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth ). (Contributed by BJ, 6-Oct-2018)

	Ref	Expression
Assertion	bj-2uplth	⊢ ( ⦅ 𝐴 , 𝐵 ⦆ = ⦅ 𝐶 , 𝐷 ⦆ ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-pr1eq	⊢ ( ⦅ 𝐴 , 𝐵 ⦆ = ⦅ 𝐶 , 𝐷 ⦆ → pr1 ⦅ 𝐴 , 𝐵 ⦆ = pr1 ⦅ 𝐶 , 𝐷 ⦆ )
2		bj-pr21val	⊢ pr1 ⦅ 𝐴 , 𝐵 ⦆ = 𝐴
3		bj-pr21val	⊢ pr1 ⦅ 𝐶 , 𝐷 ⦆ = 𝐶
4	1 2 3	3eqtr3g	⊢ ( ⦅ 𝐴 , 𝐵 ⦆ = ⦅ 𝐶 , 𝐷 ⦆ → 𝐴 = 𝐶 )
5		bj-pr2eq	⊢ ( ⦅ 𝐴 , 𝐵 ⦆ = ⦅ 𝐶 , 𝐷 ⦆ → pr2 ⦅ 𝐴 , 𝐵 ⦆ = pr2 ⦅ 𝐶 , 𝐷 ⦆ )
6		bj-pr22val	⊢ pr2 ⦅ 𝐴 , 𝐵 ⦆ = 𝐵
7		bj-pr22val	⊢ pr2 ⦅ 𝐶 , 𝐷 ⦆ = 𝐷
8	5 6 7	3eqtr3g	⊢ ( ⦅ 𝐴 , 𝐵 ⦆ = ⦅ 𝐶 , 𝐷 ⦆ → 𝐵 = 𝐷 )
9	4 8	jca	⊢ ( ⦅ 𝐴 , 𝐵 ⦆ = ⦅ 𝐶 , 𝐷 ⦆ → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) )
10		bj-2upleq	⊢ ( 𝐴 = 𝐶 → ( 𝐵 = 𝐷 → ⦅ 𝐴 , 𝐵 ⦆ = ⦅ 𝐶 , 𝐷 ⦆ ) )
11	10	imp	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ⦅ 𝐴 , 𝐵 ⦆ = ⦅ 𝐶 , 𝐷 ⦆ )
12	9 11	impbii	⊢ ( ⦅ 𝐴 , 𝐵 ⦆ = ⦅ 𝐶 , 𝐷 ⦆ ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) )