Theorem bdnel 14166

Description: Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdnel.1	|- BOUNDED A

Assertion

Ref	Expression
bdnel	|- BOUNDED x e/ A

Distinct variable group:   x, A

Proof of Theorem bdnel

Step	Hyp	Ref	Expression
1		bdnel.1	. . . 4 |- BOUNDED A
2	1	bdeli 14158	. . 3 |- BOUNDED x e. A
3	2	ax-bdn 14129	. 2 |- BOUNDED -. x e. A
4		df-nel 2441	. 2 |- ( x e/ A <-> -. x e. A )
5	3, 4	bd0r 14137	1 |- BOUNDED x e/ A

Syntax hints:   -. wn 3   e. wcel 2146   e/ wnel 2440  BOUNDED wbd 14124  BOUNDED wbdc 14152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-4 1508  ax-bd0 14125  ax-bdn 14129

This theorem depends on definitions:  df-bi 117  df-nel 2441  df-bdc 14153

This theorem is referenced by: (None)