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Theorem intunsn 4081
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 4074 . 2  |-  |^| ( A  u.  { B } )  =  (
|^| A  i^i  |^| { B } )
2 intunsn.1 . . . 4  |-  B  e. 
_V
32intsn 4078 . . 3  |-  |^| { B }  =  B
43ineq2i 3531 . 2  |-  ( |^| A  i^i  |^| { B }
)  =  ( |^| A  i^i  B )
51, 4eqtri 2455 1  |-  |^| ( A  u.  { B } )  =  (
|^| A  i^i  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948    u. cun 3310    i^i cin 3311   {csn 3806   |^|cint 4042
This theorem is referenced by:  fiint  7375  incexclem  12608  heibor1lem  26509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-un 3317  df-in 3319  df-sn 3812  df-pr 3813  df-int 4043
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