Theorem treq

Description: Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993)

	Ref	Expression
Assertion	treq	⊢ ( 𝐴 = 𝐵 → ( Tr 𝐴 ↔ Tr 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		unieq	⊢ ( 𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵 )
2	1	sseq1d	⊢ ( 𝐴 = 𝐵 → ( ∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐴 ) )
3		sseq2	⊢ ( 𝐴 = 𝐵 → ( ∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵 ) )
4	2 3	bitrd	⊢ ( 𝐴 = 𝐵 → ( ∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵 ) )
5		df-tr	⊢ ( Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴 )
6		df-tr	⊢ ( Tr 𝐵 ↔ ∪ 𝐵 ⊆ 𝐵 )
7	4 5 6	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( Tr 𝐴 ↔ Tr 𝐵 ) )