Theorem onelini 4482

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 4481	. 2 |- ( B e. A -> B C_ A )
3		dfss 3182	. 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 1373   e. wcel 2177   i^i cin 3167   C_ wss 3168   Oncon0 4415

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-in 3174  df-ss 3181  df-uni 3854  df-tr 4148  df-iord 4418  df-on 4420

This theorem is referenced by: (None)