Theorem onelssi 4346

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 4304	. 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 1480   C_ wss 3066   Oncon0 4280

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285

This theorem is referenced by:  onelini  4347  oneluni  4348  omp1eomlem  6972  enumctlemm  6992  ennnfonelemdc  11901  ennnfonelemg  11905  ctinfom  11930  isomninnlem  13214