Theorem bdcab 15341

Description: A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)

Hypothesis

Ref	Expression
bdcab.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdcab	⊢ BOUNDED {𝑥 ∣ 𝜑}

Proof of Theorem bdcab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdcab.1	. . 3 ⊢ BOUNDED 𝜑
2	1	bdab 15330	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 15339	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2179  BOUNDED wbd 15304  BOUNDED wbdc 15332

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 1460  ax-bd0 15305  ax-bdsb 15314

This theorem depends on definitions:  df-bi 117  df-clab 2180  df-bdc 15333

This theorem is referenced by:  bds  15343  bdcrab  15344  bdccsb  15352  bdcdif  15353  bdcun  15354  bdcin  15355  bdcpw  15361  bdcsn  15362  bdcuni  15368  bdcint  15369  bdciun  15370  bdciin  15371  bdcriota  15375  bj-bdfindis  15439