Theorem onelss 4484

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 4472	. 2 |- ( A e. On -> Ord A )
2		ordelss 4476	. . 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 3200   Ord word 4459   Oncon0 4460

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465

This theorem is referenced by:  onelssi  4526  ssorduni  4585  onsucelsucr  4606  tfisi  4685  tfrlem9  6484  nntri2or2  6665  phpelm  7052  exmidontri2or  7460  nninfctlemfo  12610  ennnfonelemk  13020