Theorem ordelss 4411

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 4410	. 2 |- ( Ord A -> Tr A )
2		trss 4137	. . 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 2164   C_ wss 3154   Tr wtr 4128   Ord word 4394

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 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-in 3160  df-ss 3167  df-uni 3837  df-tr 4129  df-iord 4398

This theorem is referenced by:  ordelord  4413  onelss  4419  ordsuc  4596  smores3  6348  tfrlem1  6363  tfrlemisucaccv  6380  tfrlemiubacc  6385  tfr1onlemsucaccv  6396  tfr1onlemubacc  6401  tfrcllemsucaccv  6409  tfrcllemubacc  6414  nntri1  6551  nnsseleq  6556  fict  6926  infnfi  6953  isinfinf  6955  ordiso2  7096  hashinfuni  10851