Theorem bdnel 13041

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 13033	. . 3 ⊢ BOUNDED 𝑥 ∈ 𝐴
3	2	ax-bdn 13004	. 2 ⊢ BOUNDED ¬ 𝑥 ∈ 𝐴
4		df-nel 2402	. 2 ⊢ (𝑥 ∉ 𝐴 ↔ ¬ 𝑥 ∈ 𝐴)
5	3, 4	bd0r 13012	1 ⊢ BOUNDED 𝑥 ∉ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 1480   ∉ wnel 2401  BOUNDED wbd 12999  BOUNDED wbdc 13027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-4 1487  ax-bd0 13000  ax-bdn 13004

This theorem depends on definitions:  df-bi 116  df-nel 2402  df-bdc 13028

This theorem is referenced by: (None)