Theorem ordeq 4437

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 4164	. . 3 |- ( A = B -> ( Tr A <-> Tr B ) )
2		raleq 2705	. . 3 |- ( A = B -> ( A. x e. A Tr x <-> A. x e. B Tr x ) )
3	1, 2	anbi12d 473	. 2 |- ( A = B -> ( ( Tr A /\ A. x e. A Tr x ) <-> ( Tr B /\ A. x e. B Tr x ) ) )
4		dford3 4432	. 2 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
5		dford3 4432	. 2 |- ( Ord B <-> ( Tr B /\ A. x e. B Tr x ) )
6	3, 4, 5	3bitr4g 223	1 |- ( A = B -> ( Ord A <-> Ord B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   A.wral 2486   Tr wtr 4158   Ord word 4427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159  df-iord 4431

This theorem is referenced by:  elong  4438  limeq  4442  ordelord  4446  ordtriexmidlem  4585  2ordpr  4590  issmo  6397  issmo2  6398  smoeq  6399  smores  6401  smores2  6403  smodm2  6404  smoiso  6411  tfrlem8  6427  tfri1dALT  6460