Theorem invdif 4233

Description: Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
invdif	⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Proof of Theorem invdif

Step	Hyp	Ref	Expression
1		dfin2 4225	. 2 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵)))
2		ddif 4095	. . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵
3	2	difeq2i 4077	. 2 ⊢ (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴 ∖ 𝐵)
4	1, 3	eqtri 2760	1 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1542  Vcvv 3442   ∖ cdif 3900   ∩ cin 3902

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-in 3910

This theorem is referenced by:  indif2  4235  difundi  4244  difundir  4245  difindi  4246  difindir  4247  difdif2  4250  difun1  4253  undif1  4430  difdifdir  4446  fsuppeq  8127  fsuppeqg  8128  dfsup2  9359  fsets  17108  setsdm  17109  dmxrncnvep  38637  dmcnvepres  38638  dmxrnuncnvepres  38640