Theorem trssord 4205

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 4192	. . . . . . 7 |- ( Ord B <-> ( Tr B /\ A. x e. B Tr x ) )
2	1	simprbi 269	. . . . . 6 |- ( Ord B -> A. x e. B Tr x )
3		ssralv 3085	. . . . . 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 122	. . . 4 |- ( ( A C_ B /\ Ord B ) -> A. x e. A Tr x )
6	5	anim2i 334	. . 3 |- ( ( Tr A /\ ( A C_ B /\ Ord B ) ) -> ( Tr A /\ A. x e. A Tr x ) )
7	6	3impb 1139	. 2 |- ( ( Tr A /\ A C_ B /\ Ord B ) -> ( Tr A /\ A. x e. A Tr x ) )
8		dford3 4192	. 2 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
9	7, 8	sylibr 132	1 |- ( ( Tr A /\ A C_ B /\ Ord B ) -> Ord A )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 924   A.wral 2359   C_ wss 2999   Tr wtr 3934   Ord word 4187

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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  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-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-ral 2364  df-in 3005  df-ss 3012  df-iord 4191

This theorem is referenced by:  ordelord  4206  ordin  4210  ssorduni  4302  ordtriexmidlem  4334  ordtri2or2exmidlem  4340  onsucelsucexmidlem  4343  ordsuc  4377