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Theorem onelini 4445
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 4444 . 2  |-  ( B  e.  A  ->  B  C_  A )
3 dfss 3158 . 2  |-  ( B 
C_  A  <->  B  =  ( B  i^i  A ) )
42, 3sylib 122 1  |-  ( B  e.  A  ->  B  =  ( B  i^i  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160    i^i cin 3143    C_ wss 3144   Oncon0 4378
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825  df-tr 4117  df-iord 4381  df-on 4383
This theorem is referenced by: (None)
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