Theorem bdelir 13729

Description: Inference associated with df-bdc 13723. Its converse is bdeli 13728. (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 13723	. 2 ⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
2		bdelir.1	. 2 ⊢ BOUNDED 𝑥 ∈ 𝐴
3	1, 2	mpgbir 1441	1 ⊢ BOUNDED 𝐴

Syntax hints:   ∈ wcel 2136  BOUNDED wbd 13694  BOUNDED wbdc 13722

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-gen 1437

This theorem depends on definitions:  df-bi 116  df-bdc 13723

This theorem is referenced by:  bdcv  13730  bdcab  13731  bdcvv  13739  bdcnul  13747  bdop  13757