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

Theorem intunsn 3901
Description: Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
Hypothesis
Ref Expression
intunsn.1  |-  B  e. 
_V
Assertion
Ref Expression
intunsn  |-  |^| ( A  u.  { B } )  =  (
|^| A  i^i  B
)

Proof of Theorem intunsn
StepHypRef Expression
1 intun 3894 . 2  |-  |^| ( A  u.  { B } )  =  (
|^| A  i^i  |^| { B } )
2 intunsn.1 . . . 4  |-  B  e. 
_V
32intsn 3898 . . 3  |-  |^| { B }  =  B
43ineq2i 3367 . 2  |-  ( |^| A  i^i  |^| { B }
)  =  ( |^| A  i^i  B )
51, 4eqtri 2303 1  |-  |^| ( A  u.  { B } )  =  (
|^| A  i^i  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    i^i cin 3151   {csn 3640   |^|cint 3862
This theorem is referenced by:  fiint  7133  incexclem  12295  heibor1lem  25945
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-v 2790  df-un 3157  df-in 3159  df-sn 3646  df-pr 3647  df-int 3863
  Copyright terms: Public domain W3C validator