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Theorem intunsn 3681
 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 3674 . 2 (𝐴 ∪ {𝐵}) = ( 𝐴 {𝐵})
2 intunsn.1 . . . 4 𝐵 ∈ V
32intsn 3678 . . 3 {𝐵} = 𝐵
43ineq2i 3163 . 2 ( 𝐴 {𝐵}) = ( 𝐴𝐵)
51, 4eqtri 2076 1 (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   = wceq 1259   ∈ wcel 1409  Vcvv 2574   ∪ cun 2943   ∩ cin 2944  {csn 3403  ∩ cint 3643 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-in 2952  df-sn 3409  df-pr 3410  df-int 3644 This theorem is referenced by: (None)
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