Theorem treq 2047

Description: Equality theorem for the transitive class predicate.

Assertion

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

Proof of Theorem treq

Step	Hyp	Ref	Expression
1		unieq 1927	. . . 4 ⊢ (A = B → ∪A = ∪B)
2	1	sseq1d 1527	. . 3 ⊢ (A = B → (∪A ⊆ A ↔ ∪B ⊆ A))
3		sseq2 1522	. . 3 ⊢ (A = B → (∪B ⊆ A ↔ ∪B ⊆ B))
4	2, 3	bitrd 406	. 2 ⊢ (A = B → (∪A ⊆ A ↔ ∪B ⊆ B))
5		df-tr 2042	. 2 ⊢ (Tr A ↔ ∪A ⊆ A)
6		df-tr 2042	. 2 ⊢ (Tr B ↔ ∪B ⊆ B)
7	4, 5, 6	3bitr4g 428	1 ⊢ (A = B → (Tr A ↔ Tr B))

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ⊆ wss 1487  ∪cuni 1919  Tr wtr 2041

This theorem is referenced by:  ordeq 2206  trcl 3489  tz9.1 3490  r1tr 3498

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-uni 1920  df-tr 2042

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