Theorem onelini 4360

Description: An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)

Hypothesis

Ref	Expression
on.1	|- A e. On

Assertion

Ref	Expression
onelini	|- ( B e. A -> B = ( B i^i A ) )

Proof of Theorem onelini

Step	Hyp	Ref	Expression
1		on.1	. . 3 |- A e. On
2	1	onelssi 4359	. 2 |- ( B e. A -> B C_ A )
3		dfss 3090	. 2 |- ( B C_ A <-> B = ( B i^i A ) )
4	2, 3	sylib 121	1 |- ( B e. A -> B = ( B i^i A ) )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   i^i cin 3075   C_ wss 3076   Oncon0 4293

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-uni 3745  df-tr 4035  df-iord 4296  df-on 4298

This theorem is referenced by: (None)