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Theorem invdif 4285
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 4277 . 2 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵)))
2 ddif 4151 . . 3 (V ∖ (V ∖ 𝐵)) = 𝐵
32difeq2i 4133 . 2 (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴𝐵)
41, 3eqtri 2763 1 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3478  cdif 3960  cin 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970
This theorem is referenced by:  indif2  4287  difundi  4296  difundir  4297  difindi  4298  difindir  4299  difdif2  4302  difun1  4305  undif1  4482  difdifdir  4498  fsuppeq  8199  fsuppeqg  8200  dfsup2  9482  fsets  17203  setsdm  17204
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