Theorem onelss 4365

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 4353	. 2 |- ( A e. On -> Ord A )
2		ordelss 4357	. . 3 |- ( ( Ord A /\ B e. A ) -> B C_ A )
3	2	ex 114	. 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 2136   C_ wss 3116   Ord word 4340   Oncon0 4341

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346

This theorem is referenced by:  onelssi  4407  ssorduni  4464  onsucelsucr  4485  tfisi  4564  tfrlem9  6287  nntri2or2  6466  phpelm  6832  exmidontri2or  7199  ennnfonelemk  12333