Theorem treq 4193

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 3902	. . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
2	1	sseq1d 3256	. . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐴))
3		sseq2 3251	. . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵))
4	2, 3	bitrd 188	. 2 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵))
5		df-tr 4188	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
6		df-tr 4188	. 2 ⊢ (Tr 𝐵 ↔ ∪ 𝐵 ⊆ 𝐵)
7	4, 5, 6	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397   ⊆ wss 3200  ∪ cuni 3893  Tr wtr 4187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188

This theorem is referenced by:  truni  4201  ordeq  4469  ordsucim  4598  ordom  4705  exmidonfinlem  7403