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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdcun | GIF version |
Description: The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
Ref | Expression |
---|---|
bdcdif.1 | ⊢ BOUNDED 𝐴 |
bdcdif.2 | ⊢ BOUNDED 𝐵 |
Ref | Expression |
---|---|
bdcun | ⊢ BOUNDED (𝐴 ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdcdif.1 | . . . . 5 ⊢ BOUNDED 𝐴 | |
2 | 1 | bdeli 13215 | . . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴 |
3 | bdcdif.2 | . . . . 5 ⊢ BOUNDED 𝐵 | |
4 | 3 | bdeli 13215 | . . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐵 |
5 | 2, 4 | ax-bdor 13185 | . . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) |
6 | 5 | bdcab 13218 | . 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} |
7 | df-un 3080 | . 2 ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} | |
8 | 6, 7 | bdceqir 13213 | 1 ⊢ BOUNDED (𝐴 ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 698 ∈ wcel 1481 {cab 2126 ∪ cun 3074 BOUNDED wbdc 13209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 ax-ext 2122 ax-bd0 13182 ax-bdor 13185 ax-bdsb 13191 |
This theorem depends on definitions: df-bi 116 df-clab 2127 df-cleq 2133 df-clel 2136 df-un 3080 df-bdc 13210 |
This theorem is referenced by: bdcpr 13240 bdctp 13241 bdcsuc 13249 |
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