Theorem ordeq 4199

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 3942	. . 3 |- ( A = B -> ( Tr A <-> Tr B ) )
2		raleq 2562	. . 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 4194	. 2 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
5		dford3 4194	. 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 1289   A.wral 2359   Tr wtr 3936   Ord word 4189

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193

This theorem is referenced by:  elong  4200  limeq  4204  ordelord  4208  ordtriexmidlem  4336  2ordpr  4340  issmo  6053  issmo2  6054  smoeq  6055  smores  6057  smores2  6059  smodm2  6060  smoiso  6067  tfrlem8  6083  tfri1dALT  6116