Theorem bdcdif 10652

Description: The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bdcdif.1	⊢ BOUNDED 𝐴
bdcdif.2	⊢ BOUNDED 𝐵

Assertion

Ref	Expression
bdcdif	⊢ BOUNDED (𝐴 ∖ 𝐵)

Proof of Theorem bdcdif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdcdif.1	. . . . 5 ⊢ BOUNDED 𝐴
2	1	bdeli 10637	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcdif.2	. . . . . 6 ⊢ BOUNDED 𝐵
4	3	bdeli 10637	. . . . 5 ⊢ BOUNDED 𝑥 ∈ 𝐵
5	4	ax-bdn 10608	. . . 4 ⊢ BOUNDED ¬ 𝑥 ∈ 𝐵
6	2, 5	ax-bdan 10606	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)
7	6	bdcab 10640	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
8		df-dif 2975	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
9	7, 8	bdceqir 10635	1 ⊢ BOUNDED (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 102   ∈ wcel 1433  {cab 2067   ∖ cdif 2970  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063  ax-bd0 10604  ax-bdan 10606  ax-bdn 10608  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-clab 2068  df-cleq 2074  df-clel 2077  df-dif 2975  df-bdc 10632

This theorem is referenced by:  bdcnulALT  10657