Theorem ordelss 4301

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 4300	. 2 |- ( Ord A -> Tr A )
2		trss 4035	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
3	2	imp 123	. 2 |- ( ( Tr A /\ B e. A ) -> B C_ A )
4	1, 3	sylan 281	1 |- ( ( Ord A /\ B e. A ) -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  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-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288

This theorem is referenced by:  ordelord  4303  onelss  4309  ordsuc  4478  smores3  6190  tfrlem1  6205  tfrlemisucaccv  6222  tfrlemiubacc  6227  tfr1onlemsucaccv  6238  tfr1onlemubacc  6243  tfrcllemsucaccv  6251  tfrcllemubacc  6256  nntri1  6392  nnsseleq  6397  fict  6762  infnfi  6789  isinfinf  6791  ordiso2  6920  hashinfuni  10523