Theorem neli 2509

Description: Inference associated with df-nel 2508. (Contributed by BJ, 7-Jul-2018.)

Hypothesis

Ref	Expression
neli.1	|- A e/ B

Assertion

Ref	Expression
neli	|- -. A e. B

Proof of Theorem neli

Step	Hyp	Ref	Expression
1		neli.1	. 2 |- A e/ B
2		df-nel 2508	. 2 |- ( A e/ B <-> -. A e. B )
3	1, 2	mpbi 145	1 |- -. A e. B

Syntax hints:   -. wn 3   e. wcel 2203   e/ wnel 2507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117  df-nel 2508

This theorem is referenced by:  renepnf  8321  renemnf  8322  nn0nepnf  9571  xrltnr  10112  pnfnlt  10120  nltmnf  10121  inftonninf  10804  pcgcd1  13026  pc2dvds  13028  topnex  14951