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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 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) |
Colors of variables: wff set class |
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) |
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