Theorem opeldifid

Description: Ordered pair elementhood outside of the diagonal. (Contributed by Thierry Arnoux, 1-Jan-2020)

	Ref	Expression
Assertion	opeldifid	⊢ ( Rel 𝐴 → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐴 ∖ I ) ↔ ( ⟨ 𝑋 , 𝑌 ⟩ ∈ 𝐴 ∧ 𝑋 ≠ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		reldif	⊢ ( Rel 𝐴 → Rel ( 𝐴 ∖ I ) )
2		brrelex2	⊢ ( ( Rel ( 𝐴 ∖ I ) ∧ 𝑋 ( 𝐴 ∖ I ) 𝑌 ) → 𝑌 ∈ V )
3	1 2	sylan	⊢ ( ( Rel 𝐴 ∧ 𝑋 ( 𝐴 ∖ I ) 𝑌 ) → 𝑌 ∈ V )
4		brrelex2	⊢ ( ( Rel 𝐴 ∧ 𝑋 𝐴 𝑌 ) → 𝑌 ∈ V )
5	4	adantrr	⊢ ( ( Rel 𝐴 ∧ ( 𝑋 𝐴 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑌 ∈ V )
6		brdif	⊢ ( 𝑋 ( 𝐴 ∖ I ) 𝑌 ↔ ( 𝑋 𝐴 𝑌 ∧ ¬ 𝑋 I 𝑌 ) )
7		ideqg	⊢ ( 𝑌 ∈ V → ( 𝑋 I 𝑌 ↔ 𝑋 = 𝑌 ) )
8	7	necon3bbid	⊢ ( 𝑌 ∈ V → ( ¬ 𝑋 I 𝑌 ↔ 𝑋 ≠ 𝑌 ) )
9	8	anbi2d	⊢ ( 𝑌 ∈ V → ( ( 𝑋 𝐴 𝑌 ∧ ¬ 𝑋 I 𝑌 ) ↔ ( 𝑋 𝐴 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) )
10	6 9	syl5bb	⊢ ( 𝑌 ∈ V → ( 𝑋 ( 𝐴 ∖ I ) 𝑌 ↔ ( 𝑋 𝐴 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) )
11	3 5 10	pm5.21nd	⊢ ( Rel 𝐴 → ( 𝑋 ( 𝐴 ∖ I ) 𝑌 ↔ ( 𝑋 𝐴 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) )
12		df-br	⊢ ( 𝑋 ( 𝐴 ∖ I ) 𝑌 ↔ ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐴 ∖ I ) )
13		df-br	⊢ ( 𝑋 𝐴 𝑌 ↔ ⟨ 𝑋 , 𝑌 ⟩ ∈ 𝐴 )
14	13	anbi1i	⊢ ( ( 𝑋 𝐴 𝑌 ∧ 𝑋 ≠ 𝑌 ) ↔ ( ⟨ 𝑋 , 𝑌 ⟩ ∈ 𝐴 ∧ 𝑋 ≠ 𝑌 ) )
15	11 12 14	3bitr3g	⊢ ( Rel 𝐴 → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐴 ∖ I ) ↔ ( ⟨ 𝑋 , 𝑌 ⟩ ∈ 𝐴 ∧ 𝑋 ≠ 𝑌 ) ) )