Theorem ordelss 4444

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 4443	. 2 |- ( Ord A -> Tr A )
2		trss 4167	. . 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 2178   C_ wss 3174   Tr wtr 4158   Ord word 4427

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159  df-iord 4431

This theorem is referenced by:  ordelord  4446  onelss  4452  ordsuc  4629  smores3  6402  tfrlem1  6417  tfrlemisucaccv  6434  tfrlemiubacc  6439  tfr1onlemsucaccv  6450  tfr1onlemubacc  6455  tfrcllemsucaccv  6463  tfrcllemubacc  6468  nntri1  6605  nnsseleq  6610  fict  6991  infnfi  7018  isinfinf  7020  ordiso2  7163  hashinfuni  10959