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Theorem intunsn 4915
Description: Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
Hypothesis
Ref Expression
intunsn.1 𝐵 ∈ V
Assertion
Ref Expression
intunsn (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)

Proof of Theorem intunsn
StepHypRef Expression
1 intun 4906 . 2 (𝐴 ∪ {𝐵}) = ( 𝐴 {𝐵})
2 intunsn.1 . . . 4 𝐵 ∈ V
32intsn 4912 . . 3 {𝐵} = 𝐵
43ineq2i 4139 . 2 ( 𝐴 {𝐵}) = ( 𝐴𝐵)
51, 4eqtri 2766 1 (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wcel 2111  Vcvv 3421  cun 3879  cin 3880  {csn 4556   cint 4874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rab 3071  df-v 3423  df-un 3886  df-in 3888  df-sn 4557  df-pr 4559  df-int 4875
This theorem is referenced by:  fiint  8973  incexclem  15428  heibor1lem  35734
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