Theorem treq 3990

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 3709	. . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
2	1	sseq1d 3090	. . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐴))
3		sseq2 3085	. . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵))
4	2, 3	bitrd 187	. 2 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵))
5		df-tr 3985	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
6		df-tr 3985	. 2 ⊢ (Tr 𝐵 ↔ ∪ 𝐵 ⊆ 𝐵)
7	4, 5, 6	3bitr4g 222	1 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1312   ⊆ wss 3035  ∪ cuni 3700  Tr wtr 3984

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rex 2394  df-in 3041  df-ss 3048  df-uni 3701  df-tr 3985

This theorem is referenced by:  truni  3998  ordeq  4252  ordsucim  4374  ordom  4478