Theorem bdelir 15577

Description: Inference associated with df-bdc 15571. Its converse is bdeli 15576. (Contributed by BJ, 3-Oct-2019.)

Hypothesis

Ref	Expression
bdelir.1	|- BOUNDED x e. A

Assertion

Ref	Expression
bdelir	|- BOUNDED A

Distinct variable group:   x, A

Proof of Theorem bdelir

Step	Hyp	Ref	Expression
1		df-bdc 15571	. 2 |- (BOUNDED A <-> A. xBOUNDED x e. A )
2		bdelir.1	. 2 |- BOUNDED x e. A
3	1, 2	mpgbir 1467	1 |- BOUNDED A

Syntax hints:   e. wcel 2167  BOUNDED wbd 15542  BOUNDED wbdc 15570

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 15571

This theorem is referenced by:  bdcv  15578  bdcab  15579  bdcvv  15587  bdcnul  15595  bdop  15605