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Theorem ordeq 5768
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 4791 . . 3 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2 weeq2 5132 . . 3 (𝐴 = 𝐵 → ( E We 𝐴 ↔ E We 𝐵))
31, 2anbi12d 747 . 2 (𝐴 = 𝐵 → ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐵 ∧ E We 𝐵)))
4 df-ord 5764 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
5 df-ord 5764 . 2 (Ord 𝐵 ↔ (Tr 𝐵 ∧ E We 𝐵))
63, 4, 53bitr4g 303 1 (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  Tr wtr 4785   E cep 5057   We wwe 5101  Ord word 5760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-in 3614  df-ss 3621  df-uni 4469  df-tr 4786  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764
This theorem is referenced by:  elong  5769  limeq  5773  ordelord  5783  ordun  5867  ordeleqon  7030  ordsuc  7056  ordzsl  7087  issmo  7490  issmo2  7491  smoeq  7492  smores  7494  smores2  7496  smodm2  7497  smoiso  7504  tfrlem8  7525  ordtypelem5  8468  ordtypelem7  8470  oicl  8475  oieu  8485
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