Theorem bdcin 15068

Description: The intersection 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
bdcin	⊢ BOUNDED (𝐴 ∩ 𝐵)

Proof of Theorem bdcin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdcdif.1	. . . . 5 ⊢ BOUNDED 𝐴
2	1	bdeli 15051	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcdif.2	. . . . 5 ⊢ BOUNDED 𝐵
4	3	bdeli 15051	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐵
5	2, 4	ax-bdan 15020	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
6	5	bdcab 15054	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
7		df-in 3150	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
8	6, 7	bdceqir 15049	1 ⊢ BOUNDED (𝐴 ∩ 𝐵)

Syntax hints:   ∧ wa 104   ∈ wcel 2160  {cab 2175   ∩ cin 3143  BOUNDED wbdc 15045

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171  ax-bd0 15018  ax-bdan 15020  ax-bdsb 15027

This theorem depends on definitions:  df-bi 117  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-bdc 15046

This theorem is referenced by: (None)