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Theorem invdif 4197
 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 4189 . 2 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵)))
2 ddif 4066 . . 3 (V ∖ (V ∖ 𝐵)) = 𝐵
32difeq2i 4049 . 2 (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴𝐵)
41, 3eqtri 2821 1 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  Vcvv 3441   ∖ cdif 3879   ∩ cin 3881 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-dif 3885  df-in 3889 This theorem is referenced by:  indif2  4199  difundi  4208  difundir  4209  difindi  4210  difindir  4211  difdif2  4213  difun1  4216  undif1  4384  difdifdir  4397  frnsuppeq  7837  frnsuppeqg  7838  dfsup2  8907  fsets  16525  setsdm  16526
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