Theorem bdcab 13884

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 13873	. 2 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑}
3	2	bdelir 13882	1 ⊢ BOUNDED {𝑥 ∣ 𝜑}

Syntax hints:  {cab 2156  BOUNDED wbd 13847  BOUNDED wbdc 13875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1442  ax-bd0 13848  ax-bdsb 13857

This theorem depends on definitions:  df-bi 116  df-clab 2157  df-bdc 13876

This theorem is referenced by:  bds  13886  bdcrab  13887  bdccsb  13895  bdcdif  13896  bdcun  13897  bdcin  13898  bdcpw  13904  bdcsn  13905  bdcuni  13911  bdcint  13912  bdciun  13913  bdciin  13914  bdcriota  13918  bj-bdfindis  13982