Theorem onelssi 4428

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 4386	. 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 2148   C_ wss 3129   Oncon0 4362

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-uni 3810  df-tr 4101  df-iord 4365  df-on 4367

This theorem is referenced by:  onelini  4429  oneluni  4430  omp1eomlem  7089  enumctlemm  7109  ennnfonelemdc  12391  ctinfom  12420  2o01f  14597  isomninnlem  14629  iswomninnlem  14648  ismkvnnlem  14651