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Theorem treq 4064
 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 3777 . . . 4 (𝐴 = 𝐵 𝐴 = 𝐵)
21sseq1d 3153 . . 3 (𝐴 = 𝐵 → ( 𝐴𝐴 𝐵𝐴))
3 sseq2 3148 . . 3 (𝐴 = 𝐵 → ( 𝐵𝐴 𝐵𝐵))
42, 3bitrd 187 . 2 (𝐴 = 𝐵 → ( 𝐴𝐴 𝐵𝐵))
5 df-tr 4059 . 2 (Tr 𝐴 𝐴𝐴)
6 df-tr 4059 . 2 (Tr 𝐵 𝐵𝐵)
74, 5, 63bitr4g 222 1 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332   ⊆ wss 3098  ∪ cuni 3768  Tr wtr 4058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rex 2438  df-in 3104  df-ss 3111  df-uni 3769  df-tr 4059 This theorem is referenced by:  truni  4072  ordeq  4327  ordsucim  4453  ordom  4560  exmidonfinlem  7107
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