Theorem ordelss 4379

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 4378	. 2 |- ( Ord A -> Tr A )
2		trss 4110	. . 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 2148   C_ wss 3129   Tr wtr 4101   Ord word 4362

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-ss 3142  df-uni 3810  df-tr 4102  df-iord 4366

This theorem is referenced by:  ordelord  4381  onelss  4387  ordsuc  4562  smores3  6293  tfrlem1  6308  tfrlemisucaccv  6325  tfrlemiubacc  6330  tfr1onlemsucaccv  6341  tfr1onlemubacc  6346  tfrcllemsucaccv  6354  tfrcllemubacc  6359  nntri1  6496  nnsseleq  6501  fict  6867  infnfi  6894  isinfinf  6896  ordiso2  7033  hashinfuni  10756