Theorem onelss 4418

Description: An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

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

Proof of Theorem onelss

Step	Hyp	Ref	Expression
1		eloni 4406	. 2 |- ( A e. On -> Ord A )
2		ordelss 4410	. . 3 |- ( ( Ord A /\ B e. A ) -> B C_ A )
3	2	ex 115	. 2 |- ( Ord A -> ( B e. A -> B C_ A ) )
4	1, 3	syl 14	1 |- ( A e. On -> ( B e. A -> B C_ A ) )

Syntax hints:   -> wi 4   e. wcel 2164   C_ wss 3153   Ord word 4393   Oncon0 4394

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399

This theorem is referenced by:  onelssi  4460  ssorduni  4519  onsucelsucr  4540  tfisi  4619  tfrlem9  6372  nntri2or2  6551  phpelm  6922  exmidontri2or  7303  nninfctlemfo  12177  ennnfonelemk  12557