Theorem ordeq 4357

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

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		treq 4093	. . 3 |- ( A = B -> ( Tr A <-> Tr B ) )
2		raleq 2665	. . 3 |- ( A = B -> ( A. x e. A Tr x <-> A. x e. B Tr x ) )
3	1, 2	anbi12d 470	. 2 |- ( A = B -> ( ( Tr A /\ A. x e. A Tr x ) <-> ( Tr B /\ A. x e. B Tr x ) ) )
4		dford3 4352	. 2 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
5		dford3 4352	. 2 |- ( Ord B <-> ( Tr B /\ A. x e. B Tr x ) )
6	3, 4, 5	3bitr4g 222	1 |- ( A = B -> ( Ord A <-> Ord B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   A.wral 2448   Tr wtr 4087   Ord word 4347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351

This theorem is referenced by:  elong  4358  limeq  4362  ordelord  4366  ordtriexmidlem  4503  2ordpr  4508  issmo  6267  issmo2  6268  smoeq  6269  smores  6271  smores2  6273  smodm2  6274  smoiso  6281  tfrlem8  6297  tfri1dALT  6330