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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 9941 | . . . . 5 ⊢ BOUNDED 𝑦 ∈ 𝑧 | |
2 | 1 | ax-bdex 9939 | . . . 4 ⊢ BOUNDED ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 |
3 | 2 | bdcab 9969 | . . 3 ⊢ BOUNDED {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧} |
4 | df-rex 2312 | . . . . 5 ⊢ (∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 ∈ 𝑧)) | |
5 | exancom 1499 | . . . . 5 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) | |
6 | 4, 5 | bitri 173 | . . . 4 ⊢ (∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) |
7 | 6 | abbii 2153 | . . 3 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧} = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)} |
8 | 3, 7 | bdceqi 9963 | . 2 ⊢ BOUNDED {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)} |
9 | df-uni 3581 | . 2 ⊢ ∪ 𝑥 = {𝑦 ∣ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)} | |
10 | 8, 9 | bdceqir 9964 | 1 ⊢ BOUNDED ∪ 𝑥 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∃wex 1381 {cab 2026 ∃wrex 2307 ∪ cuni 3580 BOUNDED wbdc 9960 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-bd0 9933 ax-bdex 9939 ax-bdel 9941 ax-bdsb 9942 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-rex 2312 df-uni 3581 df-bdc 9961 |
This theorem is referenced by: bj-uniex2 10036 |
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