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Theorem intunsn 4441
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 4434 . 2 (𝐴 ∪ {𝐵}) = ( 𝐴 {𝐵})
2 intunsn.1 . . . 4 𝐵 ∈ V
32intsn 4438 . . 3 {𝐵} = 𝐵
43ineq2i 3768 . 2 ( 𝐴 {𝐵}) = ( 𝐴𝐵)
51, 4eqtri 2627 1 (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wcel 1975  Vcvv 3168  cun 3533  cin 3534  {csn 4120   cint 4400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-v 3170  df-un 3540  df-in 3542  df-sn 4121  df-pr 4123  df-int 4401
This theorem is referenced by:  fiint  8095  incexclem  14349  heibor1lem  32577
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