Theorem trssord 4398

Description: A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)

Assertion

Ref	Expression
trssord	|- ( ( Tr A /\ A C_ B /\ Ord B ) -> Ord A )

Proof of Theorem trssord

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dford3 4385	. . . . . . 7 |- ( Ord B <-> ( Tr B /\ A. x e. B Tr x ) )
2	1	simprbi 275	. . . . . 6 |- ( Ord B -> A. x e. B Tr x )
3		ssralv 3234	. . . . . 6 |- ( A C_ B -> ( A. x e. B Tr x -> A. x e. A Tr x ) )
4	2, 3	syl5 32	. . . . 5 |- ( A C_ B -> ( Ord B -> A. x e. A Tr x ) )
5	4	imp 124	. . . 4 |- ( ( A C_ B /\ Ord B ) -> A. x e. A Tr x )
6	5	anim2i 342	. . 3 |- ( ( Tr A /\ ( A C_ B /\ Ord B ) ) -> ( Tr A /\ A. x e. A Tr x ) )
7	6	3impb 1201	. 2 |- ( ( Tr A /\ A C_ B /\ Ord B ) -> ( Tr A /\ A. x e. A Tr x ) )
8		dford3 4385	. 2 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
9	7, 8	sylibr 134	1 |- ( ( Tr A /\ A C_ B /\ Ord B ) -> Ord A )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   A.wral 2468   C_ wss 3144   Tr wtr 4116   Ord word 4380

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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-in 3150  df-ss 3157  df-iord 4384

This theorem is referenced by:  ordelord  4399  ordin  4403  ssorduni  4504  ordtriexmidlem  4536  ordtri2or2exmidlem  4543  onsucelsucexmidlem  4546  ordsuc  4580