Theorem ordelss 4410

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 4409	. 2 |- ( Ord A -> Tr A )
2		trss 4136	. . 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 3153   Tr wtr 4127   Ord word 4393

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 3159  df-ss 3166  df-uni 3836  df-tr 4128  df-iord 4397

This theorem is referenced by:  ordelord  4412  onelss  4418  ordsuc  4595  smores3  6346  tfrlem1  6361  tfrlemisucaccv  6378  tfrlemiubacc  6383  tfr1onlemsucaccv  6394  tfr1onlemubacc  6399  tfrcllemsucaccv  6407  tfrcllemubacc  6412  nntri1  6549  nnsseleq  6554  fict  6924  infnfi  6951  isinfinf  6953  ordiso2  7094  hashinfuni  10848