Theorem elcomplg 3218

Description: Membership in class complement. (Contributed by SF, 10-Jan-2015.)

Assertion

Ref	Expression
elcomplg	|- (A e. V -> (A e. ∼ B <-> -. A e. B))

Proof of Theorem elcomplg

Step	Hyp	Ref	Expression
1		df-compl 3212	. . 3 |- ∼ B = (B &ncap B)
2	1	eleq2i 2417	. 2 |- (A e. ∼ B <-> A e. (B &ncap B))
3		elning 3217	. . 3 |- (A e. V -> (A e. (B &ncap B) <-> (A e. B -/\ A e. B)))
4		df-nan 1288	. . . 4 |- ((A e. B -/\ A e. B) <-> -. (A e. B /\ A e. B))
5		anidm 625	. . . 4 |- ((A e. B /\ A e. B) <-> A e. B)
6	4, 5	xchbinx 301	. . 3 |- ((A e. B -/\ A e. B) <-> -. A e. B)
7	3, 6	syl6bb 252	. 2 |- (A e. V -> (A e. (B &ncap B) <-> -. A e. B))
8	2, 7	syl5bb 248	1 |- (A e. V -> (A e. ∼ B <-> -. A e. B))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   -/\ wnan 1287   e. wcel 1710   &ncap cnin 3204   ∼ ccompl 3205

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 2478  df-v 2861  df-nin 3211  df-compl 3212

This theorem is referenced by:  elin  3219  elun  3220  eldif  3221  elcompl  3225  nnadjoinpw  4521  nmembers1lem3  6270