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Theorem intunsn 4321
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 4314 . 2  |-  |^| ( A  u.  { B } )  =  (
|^| A  i^i  |^| { B } )
2 intunsn.1 . . . 4  |-  B  e. 
_V
32intsn 4318 . . 3  |-  |^| { B }  =  B
43ineq2i 3697 . 2  |-  ( |^| A  i^i  |^| { B }
)  =  ( |^| A  i^i  B )
51, 4eqtri 2496 1  |-  |^| ( A  u.  { B } )  =  (
|^| A  i^i  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474    i^i cin 3475   {csn 4027   |^|cint 4282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115  df-un 3481  df-in 3483  df-sn 4028  df-pr 4030  df-int 4283
This theorem is referenced by:  fiint  7793  incexclem  13607  heibor1lem  29908
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