Theorem invdif 4202

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 4194	. 2 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵)))
2		ddif 4071	. . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵
3	2	difeq2i 4054	. 2 ⊢ (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴 ∖ 𝐵)
4	1, 3	eqtri 2766	1 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1539  Vcvv 3432   ∖ cdif 3884   ∩ cin 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894

This theorem is referenced by:  indif2  4204  difundi  4213  difundir  4214  difindi  4215  difindir  4216  difdif2  4220  difun1  4223  undif1  4409  difdifdir  4422  frnsuppeq  7991  frnsuppeqg  7992  dfsup2  9203  fsets  16870  setsdm  16871