Theorem onelssi 4528

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 4486	. 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 2201   C_ wss 3199   Oncon0 4462

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-in 3205  df-ss 3212  df-uni 3895  df-tr 4189  df-iord 4465  df-on 4467

This theorem is referenced by:  onelini  4529  oneluni  4530  omp1eomlem  7298  enumctlemm  7318  ennnfonelemdc  13043  ctinfom  13072  2o01f  16653  isomninnlem  16701  iswomninnlem  16721  ismkvnnlem  16724