Theorem onelini 4570

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 4569	. 2 |- ( B e. A -> B C_ A )
3		dfss 3234	. 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 1402   e. wcel 2209   i^i cin 3219   C_ wss 3220   Oncon0 4503

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 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 theorem 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 3931  df-tr 4225  df-iord 4506  df-on 4508

This theorem is referenced by: (None)