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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 9276 | . . . . 5 ⊢ BOUNDED y ∈ z | |
2 | 1 | ax-bdex 9274 | . . . 4 ⊢ BOUNDED ∃z ∈ x y ∈ z |
3 | 2 | bdcab 9304 | . . 3 ⊢ BOUNDED {y ∣ ∃z ∈ x y ∈ z} |
4 | df-rex 2306 | . . . . 5 ⊢ (∃z ∈ x y ∈ z ↔ ∃z(z ∈ x ∧ y ∈ z)) | |
5 | exancom 1496 | . . . . 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 2150 | . . 3 ⊢ {y ∣ ∃z ∈ x y ∈ z} = {y ∣ ∃z(y ∈ z ∧ z ∈ x)} |
8 | 3, 7 | bdceqi 9298 | . 2 ⊢ BOUNDED {y ∣ ∃z(y ∈ z ∧ z ∈ x)} |
9 | df-uni 3572 | . 2 ⊢ ∪ x = {y ∣ ∃z(y ∈ z ∧ z ∈ x)} | |
10 | 8, 9 | bdceqir 9299 | 1 ⊢ BOUNDED ∪ x |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∃wex 1378 {cab 2023 ∃wrex 2301 ∪ cuni 3571 BOUNDED wbdc 9295 |
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 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-bd0 9268 ax-bdex 9274 ax-bdel 9276 ax-bdsb 9277 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-rex 2306 df-uni 3572 df-bdc 9296 |
This theorem is referenced by: bj-uniex2 9371 |
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