Theorem neli 2511

Description: Inference associated with df-nel 2510. (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 2510	. 2 |- ( A e/ B <-> -. A e. B )
3	1, 2	mpbi 145	1 |- -. A e. B

Syntax hints:   -. wn 3   e. wcel 2205   e/ wnel 2509

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 2510

This theorem is referenced by:  renepnf  8337  renemnf  8338  nn0nepnf  9588  xrltnr  10131  pnfnlt  10139  nltmnf  10140  inftonninf  10828  pcgcd1  13051  pc2dvds  13053  topnex  15077