Theorem bdcun 13897

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 13881	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcdif.2	. . . . 5 ⊢ BOUNDED 𝐵
4	3	bdeli 13881	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐵
5	2, 4	ax-bdor 13851	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)
6	5	bdcab 13884	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
7		df-un 3125	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
8	6, 7	bdceqir 13879	1 ⊢ BOUNDED (𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 703   ∈ wcel 2141  {cab 2156   ∪ cun 3119  BOUNDED wbdc 13875

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152  ax-bd0 13848  ax-bdor 13851  ax-bdsb 13857

This theorem depends on definitions:  df-bi 116  df-clab 2157  df-cleq 2163  df-clel 2166  df-un 3125  df-bdc 13876

This theorem is referenced by:  bdcpr  13906  bdctp  13907  bdcsuc  13915