Theorem ordelss 4394

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 4393	. 2 |- ( Ord A -> Tr A )
2		trss 4125	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
3	2	imp 124	. 2 |- ( ( Tr A /\ B e. A ) -> B C_ A )
4	1, 3	sylan 283	1 |- ( ( Ord A /\ B e. A ) -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2160   C_ wss 3144   Tr wtr 4116   Ord word 4377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825  df-tr 4117  df-iord 4381

This theorem is referenced by:  ordelord  4396  onelss  4402  ordsuc  4577  smores3  6312  tfrlem1  6327  tfrlemisucaccv  6344  tfrlemiubacc  6349  tfr1onlemsucaccv  6360  tfr1onlemubacc  6365  tfrcllemsucaccv  6373  tfrcllemubacc  6378  nntri1  6515  nnsseleq  6520  fict  6886  infnfi  6913  isinfinf  6915  ordiso2  7052  hashinfuni  10775