Theorem ordelss 4357

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 4356	. 2 |- ( Ord A -> Tr A )
2		trss 4089	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
3	2	imp 123	. 2 |- ( ( Tr A /\ B e. A ) -> B C_ A )
4	1, 3	sylan 281	1 |- ( ( Ord A /\ B e. A ) -> B C_ A )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   C_ wss 3116   Tr wtr 4080   Ord word 4340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344

This theorem is referenced by:  ordelord  4359  onelss  4365  ordsuc  4540  smores3  6261  tfrlem1  6276  tfrlemisucaccv  6293  tfrlemiubacc  6298  tfr1onlemsucaccv  6309  tfr1onlemubacc  6314  tfrcllemsucaccv  6322  tfrcllemubacc  6327  nntri1  6464  nnsseleq  6469  fict  6834  infnfi  6861  isinfinf  6863  ordiso2  7000  hashinfuni  10690