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Theorem intunsn 3894
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 3887 . 2 (𝐴 ∪ {𝐵}) = ( 𝐴 {𝐵})
2 intunsn.1 . . . 4 𝐵 ∈ V
32intsn 3891 . . 3 {𝐵} = 𝐵
43ineq2i 3345 . 2 ( 𝐴 {𝐵}) = ( 𝐴𝐵)
51, 4eqtri 2208 1 (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1363  wcel 2158  Vcvv 2749  cun 3139  cin 3140  {csn 3604   cint 3856
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-un 3145  df-in 3147  df-sn 3610  df-pr 3611  df-int 3857
This theorem is referenced by:  fiintim  6942
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