Theorem bdcin 11754

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 11737	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcdif.2	. . . . 5 ⊢ BOUNDED 𝐵
4	3	bdeli 11737	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐵
5	2, 4	ax-bdan 11706	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
6	5	bdcab 11740	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
7		df-in 3005	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
8	6, 7	bdceqir 11735	1 ⊢ BOUNDED (𝐴 ∩ 𝐵)

Syntax hints:   ∧ wa 102   ∈ wcel 1438  {cab 2074   ∩ cin 2998  BOUNDED wbdc 11731

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070  ax-bd0 11704  ax-bdan 11706  ax-bdsb 11713

This theorem depends on definitions:  df-bi 115  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-bdc 11732

This theorem is referenced by: (None)