Theorem bdel 15645

Description: The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)

Assertion

Ref	Expression
bdel	⊢ (BOUNDED 𝐴 → BOUNDED 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem bdel

Step	Hyp	Ref	Expression
1		df-bdc 15641	. 2 ⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
2		sp 1533	. 2 ⊢ (∀𝑥BOUNDED 𝑥 ∈ 𝐴 → BOUNDED 𝑥 ∈ 𝐴)
3	1, 2	sylbi 121	1 ⊢ (BOUNDED 𝐴 → BOUNDED 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1370   ∈ wcel 2175  BOUNDED wbd 15612  BOUNDED wbdc 15640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-4 1532

This theorem depends on definitions:  df-bi 117  df-bdc 15641

This theorem is referenced by:  bdeli  15646