Theorem bdelir 15503

Description: Inference associated with df-bdc 15497. Its converse is bdeli 15502. (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 15497	. 2 ⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
2		bdelir.1	. 2 ⊢ BOUNDED 𝑥 ∈ 𝐴
3	1, 2	mpgbir 1467	1 ⊢ BOUNDED 𝐴

Syntax hints:   ∈ wcel 2167  BOUNDED wbd 15468  BOUNDED wbdc 15496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1463

This theorem depends on definitions:  df-bi 117  df-bdc 15497

This theorem is referenced by:  bdcv  15504  bdcab  15505  bdcvv  15513  bdcnul  15521  bdop  15531