Theorem invdif 4240

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 4232	. 2 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ (V ∖ 𝐵)))
2		ddif 4103	. . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵
3	2	difeq2i 4086	. 2 ⊢ (𝐴 ∖ (V ∖ (V ∖ 𝐵))) = (𝐴 ∖ 𝐵)
4	1, 3	eqtri 2792	1 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1567  Vcvv 3463   ∖ cdif 3910   ∩ cin 3912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-in 3920

This theorem is referenced by:  indif2  4242  difundi  4251  difundir  4252  difindi  4253  difindir  4254  difdif2  4257  difun1  4260  undif1  4439  difdifdir  4454  fsuppeq  8167  fsuppeqg  8168  dfsup2  9400  fsets  17225  setsdm  17226  dmxrncnvep  38923  dmcnvepres  38924  dmxrnuncnvepres  38926