MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onelini Unicode version

Theorem onelini 4504
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 4501 . 2  |-  ( B  e.  A  ->  B  C_  A )
3 dfss 3167 . 2  |-  ( B 
C_  A  <->  B  =  ( B  i^i  A ) )
42, 3sylib 188 1  |-  ( B  e.  A  ->  B  =  ( B  i^i  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   Oncon0 4392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
  Copyright terms: Public domain W3C validator