Theorem bdcdif 16870

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 16855	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcdif.2	. . . . . 6 ⊢ BOUNDED 𝐵
4	3	bdeli 16855	. . . . 5 ⊢ BOUNDED 𝑥 ∈ 𝐵
5	4	ax-bdn 16826	. . . 4 ⊢ BOUNDED ¬ 𝑥 ∈ 𝐵
6	2, 5	ax-bdan 16824	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)
7	6	bdcab 16858	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
8		df-dif 3222	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
9	7, 8	bdceqir 16853	1 ⊢ BOUNDED (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 104   ∈ wcel 2209  {cab 2224   ∖ cdif 3217  BOUNDED wbdc 16849

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 16822  ax-bdan 16824  ax-bdn 16826  ax-bdsb 16831

This theorem depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-dif 3222  df-bdc 16850

This theorem is referenced by:  bdcnulALT  16875