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Theorem intunsn 4651
 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 4644 . 2 (𝐴 ∪ {𝐵}) = ( 𝐴 {𝐵})
2 intunsn.1 . . . 4 𝐵 ∈ V
32intsn 4648 . . 3 {𝐵} = 𝐵
43ineq2i 3962 . 2 ( 𝐴 {𝐵}) = ( 𝐴𝐵)
51, 4eqtri 2793 1 (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631   ∈ wcel 2145  Vcvv 3351   ∪ cun 3721   ∩ cin 3722  {csn 4317  ∩ cint 4612 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-un 3728  df-in 3730  df-sn 4318  df-pr 4320  df-int 4613 This theorem is referenced by:  fiint  8397  incexclem  14775  heibor1lem  33940
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