Theorem invdif 4207

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 4199	. 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 2762	1 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1547  Vcvv 3431   ∖ cdif 3880   ∩ cin 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-in 3890

This theorem is referenced by:  indif2  4209  difundi  4218  difundir  4219  difindi  4220  difindir  4221  difdif2  4224  difun1  4227  undif1  4404  difdifdir  4419  fsuppeq  8115  fsuppeqg  8116  dfsup2  9347  fsets  17130  setsdm  17131  dmxrncnvep  38756  dmcnvepres  38757  dmxrnuncnvepres  38759