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Theorem intunsn 3709
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 3702 . 2 (𝐴 ∪ {𝐵}) = ( 𝐴 {𝐵})
2 intunsn.1 . . . 4 𝐵 ∈ V
32intsn 3706 . . 3 {𝐵} = 𝐵
43ineq2i 3187 . 2 ( 𝐴 {𝐵}) = ( 𝐴𝐵)
51, 4eqtri 2105 1 (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1287  wcel 1436  Vcvv 2615  cun 2986  cin 2987  {csn 3431   cint 3671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-un 2992  df-in 2994  df-sn 3437  df-pr 3438  df-int 3672
This theorem is referenced by: (None)
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