Theorem invdif 3497

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 3492	. 2 ⊢ (A ∩ (V ∖ B)) = (A ∖ (V ∖ (V ∖ B)))
2		ddif 3399	. . 3 ⊢ (V ∖ (V ∖ B)) = B
3	2	difeq2i 3383	. 2 ⊢ (A ∖ (V ∖ (V ∖ B))) = (A ∖ B)
4	1, 3	eqtri 2373	1 ⊢ (A ∩ (V ∖ B)) = (A ∖ B)

Syntax hints:   = wceq 1642  Vcvv 2860   ∖ cdif 3207   ∩ cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216

This theorem is referenced by:  indif2  3499  difundi  3508  difundir  3509  difindi  3510  difindir  3511  difun1  3515  undif1  3626  difdifdir  3638