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Theorem intunsn 3882
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 3875 . 2 (𝐴 ∪ {𝐵}) = ( 𝐴 {𝐵})
2 intunsn.1 . . . 4 𝐵 ∈ V
32intsn 3879 . . 3 {𝐵} = 𝐵
43ineq2i 3333 . 2 ( 𝐴 {𝐵}) = ( 𝐴𝐵)
51, 4eqtri 2198 1 (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wcel 2148  Vcvv 2737  cun 3127  cin 3128  {csn 3592   cint 3844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-sn 3598  df-pr 3599  df-int 3845
This theorem is referenced by:  fiintim  6927
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