Theorem onelss 4532

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 4520	. 2 |- ( A e. On -> Ord A )
2		ordelss 4524	. . 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 ) )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   C_ wss 3220   Ord word 4507   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:  onelssi  4574  ssorduni  4634  onsucelsucr  4655  tfisi  4734  tfrlem9  6590  nntri2or2  6771  phpelm  7168  exmidontri2or  7602  nninfctlemfo  12817  ennnfonelemk  13291