Theorem onelini 4424

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 4423	. 2 |- ( B e. A -> B C_ A )
3		dfss 3141	. 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 1353   e. wcel 2146   i^i cin 3126   C_ wss 3127   Oncon0 4357

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-in 3133  df-ss 3140  df-uni 3806  df-tr 4097  df-iord 4360  df-on 4362

This theorem is referenced by: (None)