Theorem ordeq 6313

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 5203	. . 3 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2		weeq2 5602	. . 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 6309	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
5		df-ord 6309	. 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 5196   E cep 5513   We wwe 5566  Ord word 6305

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 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-ss 3914  df-uni 4857  df-tr 5197  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6309

This theorem is referenced by:  elong  6314  limeq  6318  ordelord  6328  ordun  6412  ordeleqon  7715  ordsuc  7744  ordzsl  7775  issmo  8268  issmo2  8269  smoeq  8270  smores  8272  smores2  8274  smodm2  8275  smoiso  8282  tfrlem8  8303  ord3  8400  ordtypelem5  9408  ordtypelem7  9410  oicl  9415  oieu  9425  fineqvnttrclse  35144  dfsucon  43626