Theorem bdcdif 17092

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 17077	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcdif.2	. . . . . 6 ⊢ BOUNDED 𝐵
4	3	bdeli 17077	. . . . 5 ⊢ BOUNDED 𝑥 ∈ 𝐵
5	4	ax-bdn 17048	. . . 4 ⊢ BOUNDED ¬ 𝑥 ∈ 𝐵
6	2, 5	ax-bdan 17046	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)
7	6	bdcab 17080	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
8		df-dif 3222	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
9	7, 8	bdceqir 17075	1 ⊢ BOUNDED (𝐴 ∖ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 104   ∈ wcel 2209  {cab 2224   ∖ cdif 3217  BOUNDED wbdc 17071

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 17044  ax-bdan 17046  ax-bdn 17048  ax-bdsb 17053

This proof depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-dif 3222  df-bdc 17072

This theorem is used by:  bdcnulALT  17097