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Theorem intunsn 4155
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 4148 . 2  |-  |^| ( A  u.  { B } )  =  (
|^| A  i^i  |^| { B } )
2 intunsn.1 . . . 4  |-  B  e. 
_V
32intsn 4152 . . 3  |-  |^| { B }  =  B
43ineq2i 3537 . 2  |-  ( |^| A  i^i  |^| { B }
)  =  ( |^| A  i^i  B )
51, 4eqtri 2453 1  |-  |^| ( A  u.  { B } )  =  (
|^| A  i^i  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1362    e. wcel 1755   _Vcvv 2962    u. cun 3314    i^i cin 3315   {csn 3865   |^|cint 4116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ral 2710  df-v 2964  df-un 3321  df-in 3323  df-sn 3866  df-pr 3868  df-int 4117
This theorem is referenced by:  fiint  7576  incexclem  13282  heibor1lem  28552
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