Theorem treq 5217

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 4878	. . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
2	1	sseq1d 3975	. . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐴))
3		sseq2 3970	. . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵))
4	2, 3	bitrd 279	. 2 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵))
5		df-tr 5210	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
6		df-tr 5210	. 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 3911  ∪ cuni 4867  Tr wtr 5209

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 3446  df-ss 3928  df-uni 4868  df-tr 5210

This theorem is referenced by:  truni  5225  trint  5227  ordeq  6327  trcl  9657  tz9.1  9658  tz9.1c  9659  tctr  9669  tcmin  9670  tc2  9671  r1tr  9705  r1elssi  9734  tcrank  9813  iswun  10633  tskr1om2  10697  elgrug  10721  grutsk  10751  dfon2lem1  35764  dfon2lem3  35766  dfon2lem4  35767  dfon2lem5  35768  dfon2lem6  35769  dfon2lem7  35770  dfon2lem8  35771  dfon2  35773  dford3lem1  43008  dford3lem2  43009  nadd1rabtr  43370  wfaxext  44976  wfaxrep  44977  wfaxpow  44980  wfaxinf2  44984  wfac8prim  44985