Theorem bdel 17037

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 17033	. 2 ⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
2		sp 1564	. 2 ⊢ (∀𝑥BOUNDED 𝑥 ∈ 𝐴 → BOUNDED 𝑥 ∈ 𝐴)
3	1, 2	sylbi 121	1 ⊢ (BOUNDED 𝐴 → BOUNDED 𝑥 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4  ∀wal 1400   ∈ wcel 2209  BOUNDED wbd 17004  BOUNDED wbdc 17032

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 17033

This theorem is used by:  bdeli  17038