Theorem ordeq 6324

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 5212	. . 3 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2		weeq2 5612	. . 3 ⊢ (𝐴 = 𝐵 → ( E We 𝐴 ↔ E We 𝐵))
3	1, 2	anbi12d 632	. 2 ⊢ (𝐴 = 𝐵 → ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐵 ∧ E We 𝐵)))
4		df-ord 6320	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
5		df-ord 6320	. 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 1541  Tr wtr 5205   E cep 5523   We wwe 5576  Ord word 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3442  df-ss 3918  df-uni 4864  df-tr 5206  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320

This theorem is referenced by:  elong  6325  limeq  6329  ordelord  6339  ordun  6423  ordeleqon  7727  ordsuc  7756  ordzsl  7787  issmo  8280  issmo2  8281  smoeq  8282  smores  8284  smores2  8286  smodm2  8287  smoiso  8294  tfrlem8  8315  ord3  8412  ordtypelem5  9429  ordtypelem7  9431  oicl  9436  oieu  9446  fineqvnttrclse  35282  dfsucon  43785