Theorem bdeli 16991

Description: Inference associated with bdel 16990. Its converse is bdelir 16992. (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 16990	. 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 16957  BOUNDED wbdc 16985

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 16986

This theorem is used by:  bdph  16995  bdcrab  16997  bdnel  16999  bdccsb  17005  bdcdif  17006  bdcun  17007  bdcin  17008  bdss  17009  bdsnss  17018  bdciun  17023  bdciin  17024  bdinex1  17044  bj-inf2vnlem3  17117