Theorem ordelss 4380

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 4379	. 2 |- ( Ord A -> Tr A )
2		trss 4111	. . 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 3130   Tr wtr 4102   Ord word 4363

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 2740  df-in 3136  df-ss 3143  df-uni 3811  df-tr 4103  df-iord 4367

This theorem is referenced by:  ordelord  4382  onelss  4388  ordsuc  4563  smores3  6294  tfrlem1  6309  tfrlemisucaccv  6326  tfrlemiubacc  6331  tfr1onlemsucaccv  6342  tfr1onlemubacc  6347  tfrcllemsucaccv  6355  tfrcllemubacc  6360  nntri1  6497  nnsseleq  6502  fict  6868  infnfi  6895  isinfinf  6897  ordiso2  7034  hashinfuni  10757