Theorem ordelss 4377

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 4376	. 2 |- ( Ord A -> Tr A )
2		trss 4108	. . 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 4099   Ord word 4360

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 3809  df-tr 4100  df-iord 4364

This theorem is referenced by:  ordelord  4379  onelss  4385  ordsuc  4560  smores3  6289  tfrlem1  6304  tfrlemisucaccv  6321  tfrlemiubacc  6326  tfr1onlemsucaccv  6337  tfr1onlemubacc  6342  tfrcllemsucaccv  6350  tfrcllemubacc  6355  nntri1  6492  nnsseleq  6497  fict  6863  infnfi  6890  isinfinf  6892  ordiso2  7029  hashinfuni  10748