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Theorem intunsn 4992
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 4985 . 2 (𝐴 ∪ {𝐵}) = ( 𝐴 {𝐵})
2 intunsn.1 . . . 4 𝐵 ∈ V
32intsn 4989 . . 3 {𝐵} = 𝐵
43ineq2i 4225 . 2 ( 𝐴 {𝐵}) = ( 𝐴𝐵)
51, 4eqtri 2763 1 (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  cin 3962  {csn 4631   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-sn 4632  df-pr 4634  df-int 4952
This theorem is referenced by:  fiint  9364  fiintOLD  9365  incexclem  15869  heibor1lem  37796
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