Theorem bdnel 11402

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 11394	. . 3 ⊢ BOUNDED 𝑥 ∈ 𝐴
3	2	ax-bdn 11365	. 2 ⊢ BOUNDED ¬ 𝑥 ∈ 𝐴
4		df-nel 2351	. 2 ⊢ (𝑥 ∉ 𝐴 ↔ ¬ 𝑥 ∈ 𝐴)
5	3, 4	bd0r 11373	1 ⊢ BOUNDED 𝑥 ∉ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 1438   ∉ wnel 2350  BOUNDED wbd 11360  BOUNDED wbdc 11388

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 1445  ax-bd0 11361  ax-bdn 11365

This theorem depends on definitions:  df-bi 115  df-nel 2351  df-bdc 11389

This theorem is referenced by: (None)