Theorem neli 2517

Description: Inference associated with df-nel 2516. (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 2516	. 2 |- ( A e/ B <-> -. A e. B )
3	1, 2	mpbi 145	1 |- -. A e. B

This proof depends on syntax axioms:   -. wn 3   e. wcel 2209   e/ wnel 2515

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This proof depends on definitions:  df-bi 117  df-nel 2516

This theorem is used by:  renepnf  8373  renemnf  8374  nn0nepnf  9642  xrltnr  10191  pnfnlt  10199  nltmnf  10200  inftonninf  10892  pcgcd1  13127  pc2dvds  13129  topnex  15236