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Theorem intunsn 4993
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 4984 . 2 (𝐴 ∪ {𝐵}) = ( 𝐴 {𝐵})
2 intunsn.1 . . . 4 𝐵 ∈ V
32intsn 4990 . . 3 {𝐵} = 𝐵
43ineq2i 4209 . 2 ( 𝐴 {𝐵}) = ( 𝐴𝐵)
51, 4eqtri 2759 1 (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2105  Vcvv 3473  cun 3946  cin 3947  {csn 4628   cint 4950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-un 3953  df-in 3955  df-sn 4629  df-pr 4631  df-int 4951
This theorem is referenced by:  fiint  9330  incexclem  15789  heibor1lem  37143
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