Theorem ordeq 4135

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 3889	. . 3 |- ( A = B -> ( Tr A <-> Tr B ) )
2		raleq 2550	. . 3 |- ( A = B -> ( A. x e. A Tr x <-> A. x e. B Tr x ) )
3	1, 2	anbi12d 457	. 2 |- ( A = B -> ( ( Tr A /\ A. x e. A Tr x ) <-> ( Tr B /\ A. x e. B Tr x ) ) )
4		dford3 4130	. 2 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
5		dford3 4130	. 2 |- ( Ord B <-> ( Tr B /\ A. x e. B Tr x ) )
6	3, 4, 5	3bitr4g 221	1 |- ( A = B -> ( Ord A <-> Ord B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   A.wral 2349   Tr wtr 3883   Ord word 4125

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-in 2980  df-ss 2987  df-uni 3610  df-tr 3884  df-iord 4129

This theorem is referenced by:  elong  4136  limeq  4140  ordelord  4144  ordtriexmidlem  4271  2ordpr  4275  issmo  5937  issmo2  5938  smoeq  5939  smores  5941  smores2  5943  smodm2  5944  smoiso  5951  tfrlem8  5967  tfri1dALT  6000