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Theorem onelini 4656
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
StepHypRef Expression
1 on.1 . . 3  |-  A  e.  On
21onelssi 4653 . 2  |-  ( B  e.  A  ->  B  C_  A )
3 dfss 3299 . 2  |-  ( B 
C_  A  <->  B  =  ( B  i^i  A ) )
42, 3sylib 189 1  |-  ( B  e.  A  ->  B  =  ( B  i^i  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    i^i cin 3283    C_ wss 3284   Oncon0 4545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-v 2922  df-in 3291  df-ss 3298  df-uni 3980  df-tr 4267  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549
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