MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  treq Structured version   Visualization version   GIF version

Theorem treq 5273
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
StepHypRef Expression
1 unieq 4919 . . . 4 (𝐴 = 𝐵 𝐴 = 𝐵)
21sseq1d 4013 . . 3 (𝐴 = 𝐵 → ( 𝐴𝐴 𝐵𝐴))
3 sseq2 4008 . . 3 (𝐴 = 𝐵 → ( 𝐵𝐴 𝐵𝐵))
42, 3bitrd 278 . 2 (𝐴 = 𝐵 → ( 𝐴𝐴 𝐵𝐵))
5 df-tr 5266 . 2 (Tr 𝐴 𝐴𝐴)
6 df-tr 5266 . 2 (Tr 𝐵 𝐵𝐵)
74, 5, 63bitr4g 313 1 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wss 3948   cuni 4908  Tr wtr 5265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3955  df-ss 3965  df-uni 4909  df-tr 5266
This theorem is referenced by:  truni  5281  trint  5283  ordeq  6371  trcl  9725  tz9.1  9726  tz9.1c  9727  tctr  9737  tcmin  9738  tc2  9739  r1tr  9773  r1elssi  9802  tcrank  9881  iswun  10701  tskr1om2  10765  elgrug  10789  grutsk  10819  dfon2lem1  34830  dfon2lem3  34832  dfon2lem4  34833  dfon2lem5  34834  dfon2lem6  34835  dfon2lem7  34836  dfon2lem8  34837  dfon2  34839  dford3lem1  41853  dford3lem2  41854  nadd1rabtr  42226
  Copyright terms: Public domain W3C validator