Theorem treq 5222

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 4882	. . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
2	1	sseq1d 3978	. . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐴))
3		sseq2 3973	. . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵))
4	2, 3	bitrd 279	. 2 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵))
5		df-tr 5215	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
6		df-tr 5215	. 2 ⊢ (Tr 𝐵 ↔ ∪ 𝐵 ⊆ 𝐵)
7	4, 5, 6	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ⊆ wss 3914  ∪ cuni 4871  Tr wtr 5214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-ss 3931  df-uni 4872  df-tr 5215

This theorem is referenced by:  truni  5230  trint  5232  ordeq  6339  trcl  9681  tz9.1  9682  tz9.1c  9683  tctr  9693  tcmin  9694  tc2  9695  r1tr  9729  r1elssi  9758  tcrank  9837  iswun  10657  tskr1om2  10721  elgrug  10745  grutsk  10775  dfon2lem1  35771  dfon2lem3  35773  dfon2lem4  35774  dfon2lem5  35775  dfon2lem6  35776  dfon2lem7  35777  dfon2lem8  35778  dfon2  35780  dford3lem1  43015  dford3lem2  43016  nadd1rabtr  43377  wfaxext  44983  wfaxrep  44984  wfaxpow  44987  wfaxinf2  44991  wfac8prim  44992