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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 13190 | . . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧 | |
2 | 1 | ax-bdex 13188 | . . . 4 ⊢ BOUNDED ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 |
3 | 2 | bdcab 13218 | . . 3 ⊢ BOUNDED {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧} |
4 | df-rex 2423 | . . . . 5 ⊢ (∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 ∈ 𝑧)) | |
5 | exancom 1588 | . . . . 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 2256 | . . 3 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧} = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)} |
8 | 3, 7 | bdceqi 13212 | . 2 ⊢ BOUNDED {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)} |
9 | df-uni 3745 | . 2 ⊢ ∪ 𝑥 = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)} | |
10 | 8, 9 | bdceqir 13213 | 1 ⊢ BOUNDED ∪ 𝑥 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∃wex 1469 {cab 2126 ∃wrex 2418 ∪ cuni 3744 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-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-bd0 13182 ax-bdex 13188 ax-bdel 13190 ax-bdsb 13191 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-rex 2423 df-uni 3745 df-bdc 13210 |
This theorem is referenced by: bj-uniex2 13285 |
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