Theorem bdnel 16624

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 16616	. . 3 ⊢ BOUNDED 𝑥 ∈ 𝐴
3	2	ax-bdn 16587	. 2 ⊢ BOUNDED ¬ 𝑥 ∈ 𝐴
4		df-nel 2508	. 2 ⊢ (𝑥 ∉ 𝐴 ↔ ¬ 𝑥 ∈ 𝐴)
5	3, 4	bd0r 16595	1 ⊢ BOUNDED 𝑥 ∉ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 2203   ∉ wnel 2507  BOUNDED wbd 16582  BOUNDED wbdc 16610

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 1559  ax-bd0 16583  ax-bdn 16587

This theorem depends on definitions:  df-bi 117  df-nel 2508  df-bdc 16611

This theorem is referenced by: (None)