Theorem treq 3851

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

Assertion

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

Proof of Theorem treq

Step	Hyp	Ref	Expression
1		unieq 3580	. . . 4 ⊢ (A = B → ∪ A = ∪ B)
2	1	sseq1d 2966	. . 3 ⊢ (A = B → (∪ A ⊆ A ↔ ∪ B ⊆ A))
3		sseq2 2961	. . 3 ⊢ (A = B → (∪ B ⊆ A ↔ ∪ B ⊆ B))
4	2, 3	bitrd 177	. 2 ⊢ (A = B → (∪ A ⊆ A ↔ ∪ B ⊆ B))
5		df-tr 3846	. 2 ⊢ (Tr A ↔ ∪ A ⊆ A)
6		df-tr 3846	. 2 ⊢ (Tr B ↔ ∪ B ⊆ B)
7	4, 5, 6	3bitr4g 212	1 ⊢ (A = B → (Tr A ↔ Tr B))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ⊆ wss 2911  ∪ cuni 3571  Tr wtr 3845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846

This theorem is referenced by:  truni  3859  ordeq  4075  ordsucim  4192  ordom  4272