Theorem bdeli 17077

Description: Inference associated with bdel 17076. Its converse is bdelir 17078. (Contributed by BJ, 3-Oct-2019.)

Hypothesis

Ref	Expression
bdeli.1	⊢ BOUNDED 𝐴

Assertion

Ref	Expression
bdeli	⊢ BOUNDED 𝑥 ∈ 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem bdeli

Step	Hyp	Ref	Expression
1		bdeli.1	. 2 ⊢ BOUNDED 𝐴
2		bdel 17076	. 2 ⊢ (BOUNDED 𝐴 → BOUNDED 𝑥 ∈ 𝐴)
3	1, 2	ax-mp 5	1 ⊢ BOUNDED 𝑥 ∈ 𝐴

This proof depends on syntax axioms:   ∈ wcel 2209  BOUNDED wbd 17043  BOUNDED wbdc 17071

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-4 1563

This proof depends on definitions:  df-bi 117  df-bdc 17072

This theorem is used by:  bdph  17081  bdcrab  17083  bdnel  17085  bdccsb  17091  bdcdif  17092  bdcun  17093  bdcin  17094  bdss  17095  bdsnss  17104  bdciun  17109  bdciin  17110  bdinex1  17130  bj-inf2vnlem3  17203