Theorem bdnel 10645

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 10637	. . 3 ⊢ BOUNDED 𝑥 ∈ 𝐴
3	2	ax-bdn 10608	. 2 ⊢ BOUNDED ¬ 𝑥 ∈ 𝐴
4		df-nel 2340	. 2 ⊢ (𝑥 ∉ 𝐴 ↔ ¬ 𝑥 ∈ 𝐴)
5	3, 4	bd0r 10616	1 ⊢ BOUNDED 𝑥 ∉ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 1433   ∉ wnel 2339  BOUNDED wbd 10603  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-4 1440  ax-bd0 10604  ax-bdn 10608

This theorem depends on definitions:  df-bi 115  df-nel 2340  df-bdc 10632

This theorem is referenced by: (None)