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Theorem intunsn 4986
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 4979 . 2 (𝐴 ∪ {𝐵}) = ( 𝐴 {𝐵})
2 intunsn.1 . . . 4 𝐵 ∈ V
32intsn 4983 . . 3 {𝐵} = 𝐵
43ineq2i 4216 . 2 ( 𝐴 {𝐵}) = ( 𝐴𝐵)
51, 4eqtri 2764 1 (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  Vcvv 3479  cun 3948  cin 3949  {csn 4625   cint 4945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-un 3955  df-in 3957  df-sn 4626  df-pr 4628  df-int 4946
This theorem is referenced by:  fiint  9367  fiintOLD  9368  incexclem  15873  heibor1lem  37817
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