Theorem ordelss 4426

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 4425	. 2 |- ( Ord A -> Tr A )
2		trss 4151	. . 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 2176   C_ wss 3166   Tr wtr 4142   Ord word 4409

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4143  df-iord 4413

This theorem is referenced by:  ordelord  4428  onelss  4434  ordsuc  4611  smores3  6379  tfrlem1  6394  tfrlemisucaccv  6411  tfrlemiubacc  6416  tfr1onlemsucaccv  6427  tfr1onlemubacc  6432  tfrcllemsucaccv  6440  tfrcllemubacc  6445  nntri1  6582  nnsseleq  6587  fict  6965  infnfi  6992  isinfinf  6994  ordiso2  7137  hashinfuni  10922