Theorem ordelss 4116

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 4115	. 2 |- ( Ord A -> Tr A )
2		trss 3863	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
3	2	imp 115	. 2 |- ( ( Tr A /\ B e. A ) -> B C_ A )
4	1, 3	sylan 267	1 |- ( ( Ord A /\ B e. A ) -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   C_ wss 2917   Tr wtr 3854   Ord word 4099

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103

This theorem is referenced by:  ordelord  4118  onelss  4124  ordsuc  4287  smores3  5908  tfrlem1  5923  tfrlemisucaccv  5939  tfrlemiubacc  5944  nntri1  6074  nnsseleq  6079  ordiso2  6357