Theorem bdcdif 16516

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 16501	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcdif.2	. . . . . 6 ⊢ BOUNDED 𝐵
4	3	bdeli 16501	. . . . 5 ⊢ BOUNDED 𝑥 ∈ 𝐵
5	4	ax-bdn 16472	. . . 4 ⊢ BOUNDED ¬ 𝑥 ∈ 𝐵
6	2, 5	ax-bdan 16470	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)
7	6	bdcab 16504	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
8		df-dif 3201	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
9	7, 8	bdceqir 16499	1 ⊢ BOUNDED (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 104   ∈ wcel 2201  {cab 2216   ∖ cdif 3196  BOUNDED wbdc 16495

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-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2212  ax-bd0 16468  ax-bdan 16470  ax-bdn 16472  ax-bdsb 16477

This theorem depends on definitions:  df-bi 117  df-clab 2217  df-cleq 2223  df-clel 2226  df-dif 3201  df-bdc 16496

This theorem is referenced by:  bdcnulALT  16521