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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdcuni | GIF version |
Description: The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.) |
Ref | Expression |
---|---|
bdcuni | ⊢ BOUNDED ∪ 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-bdel 13019 | . . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧 | |
2 | 1 | ax-bdex 13017 | . . . 4 ⊢ BOUNDED ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 |
3 | 2 | bdcab 13047 | . . 3 ⊢ BOUNDED {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧} |
4 | df-rex 2422 | . . . . 5 ⊢ (∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 ∈ 𝑧)) | |
5 | exancom 1587 | . . . . 5 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) | |
6 | 4, 5 | bitri 183 | . . . 4 ⊢ (∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) |
7 | 6 | abbii 2255 | . . 3 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧} = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)} |
8 | 3, 7 | bdceqi 13041 | . 2 ⊢ BOUNDED {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)} |
9 | df-uni 3737 | . 2 ⊢ ∪ 𝑥 = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)} | |
10 | 8, 9 | bdceqir 13042 | 1 ⊢ BOUNDED ∪ 𝑥 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∃wex 1468 {cab 2125 ∃wrex 2417 ∪ cuni 3736 BOUNDED wbdc 13038 |
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 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-bd0 13011 ax-bdex 13017 ax-bdel 13019 ax-bdsb 13020 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-rex 2422 df-uni 3737 df-bdc 13039 |
This theorem is referenced by: bj-uniex2 13114 |
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