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Theorem intunsn 4293
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 4286 . 2  |-  |^| ( A  u.  { B } )  =  (
|^| A  i^i  |^| { B } )
2 intunsn.1 . . . 4  |-  B  e. 
_V
32intsn 4290 . . 3  |-  |^| { B }  =  B
43ineq2i 3662 . 2  |-  ( |^| A  i^i  |^| { B }
)  =  ( |^| A  i^i  B )
51, 4eqtri 2452 1  |-  |^| ( A  u.  { B } )  =  (
|^| A  i^i  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1438    e. wcel 1869   _Vcvv 3082    u. cun 3435    i^i cin 3436   {csn 3997   |^|cint 4253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-v 3084  df-un 3442  df-in 3444  df-sn 3998  df-pr 4000  df-int 4254
This theorem is referenced by:  fiint  7852  incexclem  13887  heibor1lem  32099
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