Theorem ordelss 4134

Description: An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)

Assertion

Ref	Expression
ordelss	|- ( ( Ord A /\ B e. A ) -> B C_ A )

Proof of Theorem ordelss

Step	Hyp	Ref	Expression
1		ordtr 4133	. 2 |- ( Ord A -> Tr A )
2		trss 3884	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
3	2	imp 122	. 2 |- ( ( Tr A /\ B e. A ) -> B C_ A )
4	1, 3	sylan 277	1 |- ( ( Ord A /\ B e. A ) -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   C_ wss 2973   Tr wtr 3875   Ord word 4117

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876  df-iord 4121

This theorem is referenced by:  ordelord  4136  onelss  4142  ordsuc  4306  smores3  5931  tfrlem1  5946  tfrlemisucaccv  5962  tfrlemiubacc  5967  nntri1  6097  nnsseleq  6102  ordiso2  6446