Theorem ordelss 4415

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 4414	. 2 |- ( Ord A -> Tr A )
2		trss 4141	. . 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 2167   C_ wss 3157   Tr wtr 4132   Ord word 4398

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-uni 3841  df-tr 4133  df-iord 4402

This theorem is referenced by:  ordelord  4417  onelss  4423  ordsuc  4600  smores3  6352  tfrlem1  6367  tfrlemisucaccv  6384  tfrlemiubacc  6389  tfr1onlemsucaccv  6400  tfr1onlemubacc  6405  tfrcllemsucaccv  6413  tfrcllemubacc  6418  nntri1  6555  nnsseleq  6560  fict  6930  infnfi  6957  isinfinf  6959  ordiso2  7102  hashinfuni  10871