Theorem bdeli 16970

Description: Inference associated with bdel 16969. Its converse is bdelir 16971. (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 16969	. 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 16936  BOUNDED wbdc 16964

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 16965

This theorem is used by:  bdph  16974  bdcrab  16976  bdnel  16978  bdccsb  16984  bdcdif  16985  bdcun  16986  bdcin  16987  bdss  16988  bdsnss  16997  bdciun  17002  bdciin  17003  bdinex1  17023  bj-inf2vnlem3  17096