Theorem bdelir 10889

Description: Inference associated with df-bdc 10883. Its converse is bdeli 10888. (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 10883	. 2 ⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
2		bdelir.1	. 2 ⊢ BOUNDED 𝑥 ∈ 𝐴
3	1, 2	mpgbir 1383	1 ⊢ BOUNDED 𝐴

Syntax hints:   ∈ wcel 1434  BOUNDED wbd 10854  BOUNDED wbdc 10882

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-gen 1379

This theorem depends on definitions:  df-bi 115  df-bdc 10883

This theorem is referenced by:  bdcv  10890  bdcab  10891  bdcvv  10899  bdcnul  10907  bdop  10917