Theorem ordeq 6402

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 5291	. . 3 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2		weeq2 5688	. . 3 ⊢ (𝐴 = 𝐵 → ( E We 𝐴 ↔ E We 𝐵))
3	1, 2	anbi12d 631	. 2 ⊢ (𝐴 = 𝐵 → ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐵 ∧ E We 𝐵)))
4		df-ord 6398	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
5		df-ord 6398	. 2 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ E We 𝐵))
6	3, 4, 5	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  Tr wtr 5283   E cep 5598   We wwe 5651  Ord word 6394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-v 3490  df-ss 3993  df-uni 4932  df-tr 5284  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398

This theorem is referenced by:  elong  6403  limeq  6407  ordelord  6417  ordun  6499  ordeleqon  7817  ordsuc  7849  ordsucOLD  7850  ordzsl  7882  issmo  8404  issmo2  8405  smoeq  8406  smores  8408  smores2  8410  smodm2  8411  smoiso  8418  tfrlem8  8440  ord3  8539  ordtypelem5  9591  ordtypelem7  9593  oicl  9598  oieu  9608  dfsucon  43485