Theorem ordeq 6320

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 5188	. . 3 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2		weeq2 5608	. . 3 ⊢ (𝐴 = 𝐵 → ( E We 𝐴 ↔ E We 𝐵))
3	1, 2	anbi12d 639	. 2 ⊢ (𝐴 = 𝐵 → ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐵 ∧ E We 𝐵)))
4		df-ord 6316	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
5		df-ord 6316	. 2 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ E We 𝐵))
6	3, 4, 5	3bitr4g 316	1 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548  Tr wtr 5181   E cep 5519   We wwe 5572  Ord word 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-v 3435  df-ss 3901  df-uni 4841  df-tr 5182  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-ord 6316

This theorem is referenced by:  elong  6321  limeq  6325  ordelord  6335  ordun  6419  ordeleqon  7728  ordsuc  7757  ordzsl  7788  issmo  8281  issmo2  8282  smoeq  8283  smores  8285  smores2  8287  smodm2  8288  smoiso  8295  tfrlem8  8317  ord3  8414  ordtypelem5  9431  ordtypelem7  9433  oicl  9438  oieu  9448  fineqvnttrclse  35318  dfsucon  43980