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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdcin | GIF version |
Description: The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
Ref | Expression |
---|---|
bdcdif.1 | ⊢ BOUNDED 𝐴 |
bdcdif.2 | ⊢ BOUNDED 𝐵 |
Ref | Expression |
---|---|
bdcin | ⊢ BOUNDED (𝐴 ∩ 𝐵) |
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) |
Colors of variables: wff set class |
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) |
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