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

Theorem treq 5144
 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 4809 . . . 4 (𝐴 = 𝐵 𝐴 = 𝐵)
21sseq1d 3923 . . 3 (𝐴 = 𝐵 → ( 𝐴𝐴 𝐵𝐴))
3 sseq2 3918 . . 3 (𝐴 = 𝐵 → ( 𝐵𝐴 𝐵𝐵))
42, 3bitrd 282 . 2 (𝐴 = 𝐵 → ( 𝐴𝐴 𝐵𝐵))
5 df-tr 5139 . 2 (Tr 𝐴 𝐴𝐴)
6 df-tr 5139 . 2 (Tr 𝐵 𝐵𝐵)
74, 5, 63bitr4g 317 1 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ⊆ wss 3858  ∪ cuni 4798  Tr wtr 5138 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3865  df-ss 3875  df-uni 4799  df-tr 5139 This theorem is referenced by:  truni  5152  trint  5154  ordeq  6176  trcl  9203  tz9.1  9204  tz9.1c  9205  tctr  9215  tcmin  9216  tc2  9217  r1tr  9238  r1elssi  9267  tcrank  9346  iswun  10164  tskr1om2  10228  elgrug  10252  grutsk  10282  dfon2lem1  33275  dfon2lem3  33277  dfon2lem4  33278  dfon2lem5  33279  dfon2lem6  33280  dfon2lem7  33281  dfon2lem8  33282  dfon2  33284  dford3lem1  40340  dford3lem2  40341
 Copyright terms: Public domain W3C validator