Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  invdif Structured version   Visualization version   GIF version

Theorem invdif 4171
 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 4163 . 2 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵)))
2 ddif 4040 . . 3 (V ∖ (V ∖ 𝐵)) = 𝐵
32difeq2i 4023 . 2 (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴𝐵)
41, 3eqtri 2821 1 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1525  Vcvv 3440   ∖ cdif 3862   ∩ cin 3864 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-dif 3868  df-in 3872 This theorem is referenced by:  indif2  4173  difundi  4182  difundir  4183  difindi  4184  difindir  4185  difdif2  4187  difun1  4190  undif1  4344  difdifdir  4357  frnsuppeq  7700  dfsup2  8761  fsets  16349  setsdm  16350
 Copyright terms: Public domain W3C validator