Theorem onelssi 4574

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 4532	. 2 |- ( A e. On -> ( B e. A -> B C_ A ) )
3	1, 2	ax-mp 5	1 |- ( B e. A -> B C_ A )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   C_ wss 3220   Oncon0 4508

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-uni 3936  df-tr 4230  df-iord 4511  df-on 4513

This theorem is used by:  onelini  4575  oneluni  4576  omp1eomlem  7434  enumctlemm  7454  ennnfonelemdc  13290  ctinfom  13319  2o01f  17024  isomninnlem  17079  iswomninnlem  17099  ismkvnnlem  17102