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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 9966 | . . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴 |
3 | bdcdif.2 | . . . . 5 ⊢ BOUNDED 𝐵 | |
4 | 3 | bdeli 9966 | . . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐵 |
5 | 2, 4 | ax-bdan 9935 | . . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) |
6 | 5 | bdcab 9969 | . 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
7 | df-in 2924 | . 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
8 | 6, 7 | bdceqir 9964 | 1 ⊢ BOUNDED (𝐴 ∩ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∈ wcel 1393 {cab 2026 ∩ cin 2916 BOUNDED wbdc 9960 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 ax-ext 2022 ax-bd0 9933 ax-bdan 9935 ax-bdsb 9942 |
This theorem depends on definitions: df-bi 110 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-bdc 9961 |
This theorem is referenced by: (None) |
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