Theorem trssord 4382

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 4369	. . . . . . 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 3221	. . . . . 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 1199	. 2 |- ( ( Tr A /\ A C_ B /\ Ord B ) -> ( Tr A /\ A. x e. A Tr x ) )
8		dford3 4369	. 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 978   A.wral 2455   C_ wss 3131   Tr wtr 4103   Ord word 4364

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-in 3137  df-ss 3144  df-iord 4368

This theorem is referenced by:  ordelord  4383  ordin  4387  ssorduni  4488  ordtriexmidlem  4520  ordtri2or2exmidlem  4527  onsucelsucexmidlem  4530  ordsuc  4564