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Theorem ordeq 5948
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 4951 . . 3 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2 weeq2 5301 . . 3 (𝐴 = 𝐵 → ( E We 𝐴 ↔ E We 𝐵))
31, 2anbi12d 625 . 2 (𝐴 = 𝐵 → ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐵 ∧ E We 𝐵)))
4 df-ord 5944 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
5 df-ord 5944 . 2 (Ord 𝐵 ↔ (Tr 𝐵 ∧ E We 𝐵))
63, 4, 53bitr4g 306 1 (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  Tr wtr 4945   E cep 5224   We wwe 5270  Ord word 5940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-in 3776  df-ss 3783  df-uni 4629  df-tr 4946  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944
This theorem is referenced by:  elong  5949  limeq  5953  ordelord  5963  ordun  6042  ordeleqon  7222  ordsuc  7248  ordzsl  7279  issmo  7684  issmo2  7685  smoeq  7686  smores  7688  smores2  7690  smodm2  7691  smoiso  7698  tfrlem8  7719  ordtypelem5  8669  ordtypelem7  8671  oicl  8676  oieu  8686
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