Theorem bdnel 15500

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 ∉ 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem bdnel

Step	Hyp	Ref	Expression
1		bdnel.1	. . . 4 ⊢ BOUNDED 𝐴
2	1	bdeli 15492	. . 3 ⊢ BOUNDED 𝑥 ∈ 𝐴
3	2	ax-bdn 15463	. 2 ⊢ BOUNDED ¬ 𝑥 ∈ 𝐴
4		df-nel 2463	. 2 ⊢ (𝑥 ∉ 𝐴 ↔ ¬ 𝑥 ∈ 𝐴)
5	3, 4	bd0r 15471	1 ⊢ BOUNDED 𝑥 ∉ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 2167   ∉ wnel 2462  BOUNDED wbd 15458  BOUNDED wbdc 15486

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 1524  ax-bd0 15459  ax-bdn 15463

This theorem depends on definitions:  df-bi 117  df-nel 2463  df-bdc 15487

This theorem is referenced by: (None)