Theorem bdelir 11738

Description: Inference associated with df-bdc 11732. Its converse is bdeli 11737. (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 11732	. 2 ⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
2		bdelir.1	. 2 ⊢ BOUNDED 𝑥 ∈ 𝐴
3	1, 2	mpgbir 1387	1 ⊢ BOUNDED 𝐴

Syntax hints:   ∈ wcel 1438  BOUNDED wbd 11703  BOUNDED wbdc 11731

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 1383

This theorem depends on definitions:  df-bi 115  df-bdc 11732

This theorem is referenced by:  bdcv  11739  bdcab  11740  bdcvv  11748  bdcnul  11756  bdop  11766