Theorem treq 4133

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 3844	. . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
2	1	sseq1d 3208	. . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐴))
3		sseq2 3203	. . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵))
4	2, 3	bitrd 188	. 2 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵))
5		df-tr 4128	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
6		df-tr 4128	. 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 1364   ⊆ wss 3153  ∪ cuni 3835  Tr wtr 4127

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-in 3159  df-ss 3166  df-uni 3836  df-tr 4128

This theorem is referenced by:  truni  4141  ordeq  4403  ordsucim  4532  ordom  4639  exmidonfinlem  7253