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Theorem invdif 4069
Description: Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
invdif (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)

Proof of Theorem invdif
StepHypRef Expression
1 dfin2 4061 . 2 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵)))
2 ddif 3940 . . 3 (V ∖ (V ∖ 𝐵)) = 𝐵
32difeq2i 3923 . 2 (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴𝐵)
41, 3eqtri 2821 1 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  Vcvv 3385  cdif 3766  cin 3768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-in 3776
This theorem is referenced by:  indif2  4071  difundi  4080  difundir  4081  difindi  4082  difindir  4083  difdif2  4085  difun1  4088  undif1  4237  difdifdir  4250  frnsuppeq  7544  dfsup2  8592  fsets  16217  setsdm  16218
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