Theorem trssord 4477

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 4464	. . . . . . 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 3291	. . . . . 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 1225	. 2 |- ( ( Tr A /\ A C_ B /\ Ord B ) -> ( Tr A /\ A. x e. A Tr x ) )
8		dford3 4464	. 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 1004   A.wral 2510   C_ wss 3200   Tr wtr 4187   Ord word 4459

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2515  df-in 3206  df-ss 3213  df-iord 4463

This theorem is referenced by:  ordelord  4478  ordin  4482  ssorduni  4585  ordtriexmidlem  4617  ordtri2or2exmidlem  4624  onsucelsucexmidlem  4627  ordsuc  4661