Theorem trssord 4302

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 4289	. . . . . . 7 |- ( Ord B <-> ( Tr B /\ A. x e. B Tr x ) )
2	1	simprbi 273	. . . . . 6 |- ( Ord B -> A. x e. B Tr x )
3		ssralv 3161	. . . . . 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 123	. . . 4 |- ( ( A C_ B /\ Ord B ) -> A. x e. A Tr x )
6	5	anim2i 339	. . 3 |- ( ( Tr A /\ ( A C_ B /\ Ord B ) ) -> ( Tr A /\ A. x e. A Tr x ) )
7	6	3impb 1177	. 2 |- ( ( Tr A /\ A C_ B /\ Ord B ) -> ( Tr A /\ A. x e. A Tr x ) )
8		dford3 4289	. 2 |- ( Ord A <-> ( Tr A /\ A. x e. A Tr x ) )
9	7, 8	sylibr 133	1 |- ( ( Tr A /\ A C_ B /\ Ord B ) -> Ord A )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   A.wral 2416   C_ wss 3071   Tr wtr 4026   Ord word 4284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-in 3077  df-ss 3084  df-iord 4288

This theorem is referenced by:  ordelord  4303  ordin  4307  ssorduni  4403  ordtriexmidlem  4435  ordtri2or2exmidlem  4441  onsucelsucexmidlem  4444  ordsuc  4478