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Theorem onelini 4322
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 4321 . 2  |-  ( B  e.  A  ->  B  C_  A )
3 dfss 3055 . 2  |-  ( B 
C_  A  <->  B  =  ( B  i^i  A ) )
42, 3sylib 121 1  |-  ( B  e.  A  ->  B  =  ( B  i^i  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465    i^i cin 3040    C_ wss 3041   Oncon0 4255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260
This theorem is referenced by: (None)
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