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Theorem invdif 3849
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 3843 . 2 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵)))
2 ddif 3725 . . 3 (V ∖ (V ∖ 𝐵)) = 𝐵
32difeq2i 3708 . 2 (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴𝐵)
41, 3eqtri 2643 1 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  Vcvv 3189  cdif 3556  cin 3558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3191  df-dif 3562  df-in 3566
This theorem is referenced by:  indif2  3851  difundi  3860  difundir  3861  difindi  3862  difindir  3863  difdif2  3865  difun1  3868  undif1  4020  difdifdir  4033  frnsuppeq  7259  dfsup2  8302  fsets  15823  setsdm  15824
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