Theorem onelss 4490

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 4478	. 2 |- ( A e. On -> Ord A )
2		ordelss 4482	. . 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 2202   C_ wss 3201   Ord word 4465   Oncon0 4466

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-in 3207  df-ss 3214  df-uni 3899  df-tr 4193  df-iord 4469  df-on 4471

This theorem is referenced by:  onelssi  4532  ssorduni  4591  onsucelsucr  4612  tfisi  4691  tfrlem9  6528  nntri2or2  6709  phpelm  7096  exmidontri2or  7504  nninfctlemfo  12674  ennnfonelemk  13084