Theorem invdif 4227

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 4219	. 2 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵)))
2		ddif 4080	. . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵
3	2	difeq2i 4063	. 2 ⊢ (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴 ∖ 𝐵)
4	1, 3	eqtri 2786	1 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3454   ∖ cdif 4048   ∩ cin 4092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-dif 4050  df-in 4094

This theorem is used by:  indif2  4229  difundi  4238  difundir  4239  difindi  4240  difindir  4241  difdif2  4244  difun1  4247  undif1  4433  difdifdir  4450  fsuppeq  8230  fsuppeqg  8231  dfsup2  9473  fsets  17391  setsdm  17392  dmxrncnvep  39587  dmcnvepres  39588  dmxrnuncnvepres  39590