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Theorem ordeq 6193
Description: Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
Assertion
Ref Expression
ordeq (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))

Proof of Theorem ordeq
StepHypRef Expression
1 treq 5171 . . 3 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2 weeq2 5539 . . 3 (𝐴 = 𝐵 → ( E We 𝐴 ↔ E We 𝐵))
31, 2anbi12d 632 . 2 (𝐴 = 𝐵 → ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐵 ∧ E We 𝐵)))
4 df-ord 6189 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
5 df-ord 6189 . 2 (Ord 𝐵 ↔ (Tr 𝐵 ∧ E We 𝐵))
63, 4, 53bitr4g 316 1 (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  Tr wtr 5165   E cep 5459   We wwe 5508  Ord word 6185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ral 3143  df-rex 3144  df-in 3943  df-ss 3952  df-uni 4833  df-tr 5166  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-ord 6189
This theorem is referenced by:  elong  6194  limeq  6198  ordelord  6208  ordun  6287  ordeleqon  7497  ordsuc  7523  ordzsl  7554  issmo  7979  issmo2  7980  smoeq  7981  smores  7983  smores2  7985  smodm2  7986  smoiso  7993  tfrlem8  8014  ordtypelem5  8980  ordtypelem7  8982  oicl  8987  oieu  8997  dfsucon  39882
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