Theorem elning 3218

Description: Membership in anti-intersection. (Contributed by SF, 10-Jan-2015.)

Assertion

Ref	Expression
elning	|- (A e. V -> (A e. (B &ncap C) <-> (A e. B -/\ A e. C)))

Proof of Theorem elning

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2413	. . 3 |- (x = A -> (x e. B <-> A e. B))
2		eleq1 2413	. . 3 |- (x = A -> (x e. C <-> A e. C))
3	1, 2	nanbi12d 1303	. 2 |- (x = A -> ((x e. B -/\ x e. C) <-> (A e. B -/\ A e. C)))
4		df-nin 3212	. 2 |- (B &ncap C) = {x | (x e. B -/\ x e. C)}
5	3, 4	elab2g 2988	1 |- (A e. V -> (A e. (B &ncap C) <-> (A e. B -/\ A e. C)))

Syntax hints:   -> wi 4   <-> wb 176   -/\ wnan 1287   = wceq 1642   e. wcel 1710   &ncap cnin 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 2479  df-v 2862  df-nin 3212

This theorem is referenced by:  elcomplg  3219  elin  3220  elun  3221  elnin  3225