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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 11692 | . . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴 |
3 | bdcdif.2 | . . . . 5 ⊢ BOUNDED 𝐵 | |
4 | 3 | bdeli 11692 | . . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐵 |
5 | 2, 4 | ax-bdor 11662 | . . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) |
6 | 5 | bdcab 11695 | . 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} |
7 | df-un 3003 | . 2 ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} | |
8 | 6, 7 | bdceqir 11690 | 1 ⊢ BOUNDED (𝐴 ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 664 ∈ wcel 1438 {cab 2074 ∪ cun 2997 BOUNDED wbdc 11686 |
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 11659 ax-bdor 11662 ax-bdsb 11668 |
This theorem depends on definitions: df-bi 115 df-clab 2075 df-cleq 2081 df-clel 2084 df-un 3003 df-bdc 11687 |
This theorem is referenced by: bdcpr 11717 bdctp 11718 bdcsuc 11726 |
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