Theorem ordeq 6166

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

Step	Hyp	Ref	Expression
1		treq 5142	. . 3 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2		weeq2 5508	. . 3 ⊢ (𝐴 = 𝐵 → ( E We 𝐴 ↔ E We 𝐵))
3	1, 2	anbi12d 633	. 2 ⊢ (𝐴 = 𝐵 → ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐵 ∧ E We 𝐵)))
4		df-ord 6162	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
5		df-ord 6162	. 2 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ E We 𝐵))
6	3, 4, 5	3bitr4g 317	1 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Tr wtr 5136   E cep 5429   We wwe 5477  Ord word 6158

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 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801  df-tr 5137  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162

This theorem is referenced by:  elong  6167  limeq  6171  ordelord  6181  ordun  6260  ordeleqon  7483  ordsuc  7509  ordzsl  7540  issmo  7968  issmo2  7969  smoeq  7970  smores  7972  smores2  7974  smodm2  7975  smoiso  7982  tfrlem8  8003  ordtypelem5  8970  ordtypelem7  8972  oicl  8977  oieu  8987  dfsucon  40231