Theorem bdcun 15475

Description: The union 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
bdcun	⊢ BOUNDED (𝐴 ∪ 𝐵)

Proof of Theorem bdcun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdcdif.1	. . . . 5 ⊢ BOUNDED 𝐴
2	1	bdeli 15459	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcdif.2	. . . . 5 ⊢ BOUNDED 𝐵
4	3	bdeli 15459	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐵
5	2, 4	ax-bdor 15429	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)
6	5	bdcab 15462	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
7		df-un 3161	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
8	6, 7	bdceqir 15457	1 ⊢ BOUNDED (𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 709   ∈ wcel 2167  {cab 2182   ∪ cun 3155  BOUNDED wbdc 15453

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15426  ax-bdor 15429  ax-bdsb 15435

This theorem depends on definitions:  df-bi 117  df-clab 2183  df-cleq 2189  df-clel 2192  df-un 3161  df-bdc 15454

This theorem is referenced by:  bdcpr  15484  bdctp  15485  bdcsuc  15493