Theorem ordelss 4470

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 4469	. 2 |- ( Ord A -> Tr A )
2		trss 4191	. . 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 2200   C_ wss 3197   Tr wtr 4182   Ord word 4453

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-ss 3210  df-uni 3889  df-tr 4183  df-iord 4457

This theorem is referenced by:  ordelord  4472  onelss  4478  ordsuc  4655  smores3  6439  tfrlem1  6454  tfrlemisucaccv  6471  tfrlemiubacc  6476  tfr1onlemsucaccv  6487  tfr1onlemubacc  6492  tfrcllemsucaccv  6500  tfrcllemubacc  6505  nntri1  6642  nnsseleq  6647  fict  7030  infnfi  7057  isinfinf  7059  ordiso2  7202  hashinfuni  10999