Theorem onelssi 4520

Description: A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)

Hypothesis

Ref	Expression
on.1	|- A e. On

Assertion

Ref	Expression
onelssi	|- ( B e. A -> B C_ A )

Proof of Theorem onelssi

Step	Hyp	Ref	Expression
1		on.1	. 2 |- A e. On
2		onelss 4478	. 2 |- ( A e. On -> ( B e. A -> B C_ A ) )
3	1, 2	ax-mp 5	1 |- ( B e. A -> B C_ A )

Syntax hints:   -> wi 4   e. wcel 2200   C_ wss 3197   Oncon0 4454

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-uni 3889  df-tr 4183  df-iord 4457  df-on 4459

This theorem is referenced by:  onelini  4521  oneluni  4522  omp1eomlem  7261  enumctlemm  7281  ennnfonelemdc  12970  ctinfom  12999  2o01f  16358  isomninnlem  16398  iswomninnlem  16417  ismkvnnlem  16420