Theorem bdcdif 13059

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 13044	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcdif.2	. . . . . 6 ⊢ BOUNDED 𝐵
4	3	bdeli 13044	. . . . 5 ⊢ BOUNDED 𝑥 ∈ 𝐵
5	4	ax-bdn 13015	. . . 4 ⊢ BOUNDED ¬ 𝑥 ∈ 𝐵
6	2, 5	ax-bdan 13013	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)
7	6	bdcab 13047	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
8		df-dif 3073	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
9	7, 8	bdceqir 13042	1 ⊢ BOUNDED (𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 103   ∈ wcel 1480  {cab 2125   ∖ cdif 3068  BOUNDED wbdc 13038

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-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121  ax-bd0 13011  ax-bdan 13013  ax-bdn 13015  ax-bdsb 13020

This theorem depends on definitions:  df-bi 116  df-clab 2126  df-cleq 2132  df-clel 2135  df-dif 3073  df-bdc 13039

This theorem is referenced by:  bdcnulALT  13064