Theorem bdelir 16885

Description: Inference associated with df-bdc 16879. Its converse is bdeli 16884. (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 16879	. 2 ⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
2		bdelir.1	. 2 ⊢ BOUNDED 𝑥 ∈ 𝐴
3	1, 2	mpgbir 1506	1 ⊢ BOUNDED 𝐴

This proof depends on syntax axioms:   ∈ wcel 2209  BOUNDED wbd 16850  BOUNDED wbdc 16878

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1502

This proof depends on definitions:  df-bi 117  df-bdc 16879

This theorem is used by:  bdcv  16886  bdcab  16887  bdcvv  16895  bdcnul  16903  bdop  16913