Theorem bdelir 9967

Description: Inference associated with df-bdc 9961. Its converse is bdeli 9966. (Contributed by BJ, 3-Oct-2019.)

Hypothesis

Ref	Expression
bdelir.1	⊢ BOUNDED 𝑥 ∈ 𝐴

Assertion

Ref	Expression
bdelir	⊢ BOUNDED 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem bdelir

Step	Hyp	Ref	Expression
1		df-bdc 9961	. 2 ⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
2		bdelir.1	. 2 ⊢ BOUNDED 𝑥 ∈ 𝐴
3	1, 2	mpgbir 1342	1 ⊢ BOUNDED 𝐴

Syntax hints:   ∈ wcel 1393  BOUNDED wbd 9932  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338

This theorem depends on definitions:  df-bi 110  df-bdc 9961

This theorem is referenced by:  bdcv  9968  bdcab  9969  bdcvv  9977  bdcnul  9985  bdop  9995