Theorem onelini 4442

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 4441	. 2 |- ( B e. A -> B C_ A )
3		dfss 3155	. 2 |- ( B C_ A <-> B = ( B i^i A ) )
4	2, 3	sylib 122	1 |- ( B e. A -> B = ( B i^i A ) )

Syntax hints:   -> wi 4   = wceq 1363   e. wcel 2158   i^i cin 3140   C_ wss 3141   Oncon0 4375

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-in 3147  df-ss 3154  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380

This theorem is referenced by: (None)