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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdreu | GIF version |
Description: Boundedness of
existential uniqueness.
Remark regarding restricted quantifiers: the formula ∀𝑥 ∈ 𝐴𝜑 need not be bounded even if 𝐴 and 𝜑 are. Indeed, V is bounded by bdcvv 13226, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 13193, if ∀𝑥 ∈ V𝜑 were bounded when 𝜑 is bounded, then ∀𝑥𝜑 would be bounded as well when 𝜑 is bounded, which is not the case. The same remark holds with ∃, ∃!, ∃*. (Contributed by BJ, 16-Oct-2019.) |
Ref | Expression |
---|---|
bdreu.1 | ⊢ BOUNDED 𝜑 |
Ref | Expression |
---|---|
bdreu | ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdreu.1 | . . . 4 ⊢ BOUNDED 𝜑 | |
2 | 1 | ax-bdex 13188 | . . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑 |
3 | ax-bdeq 13189 | . . . . . 6 ⊢ BOUNDED 𝑥 = 𝑧 | |
4 | 1, 3 | ax-bdim 13183 | . . . . 5 ⊢ BOUNDED (𝜑 → 𝑥 = 𝑧) |
5 | 4 | ax-bdal 13187 | . . . 4 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧) |
6 | 5 | ax-bdex 13188 | . . 3 ⊢ BOUNDED ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧) |
7 | 2, 6 | ax-bdan 13184 | . 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)) |
8 | reu3 2878 | . 2 ⊢ (∃!𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧))) | |
9 | 7, 8 | bd0r 13194 | 1 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wral 2417 ∃wrex 2418 ∃!wreu 2419 BOUNDED wbd 13181 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-bd0 13182 ax-bdim 13183 ax-bdan 13184 ax-bdal 13187 ax-bdex 13188 ax-bdeq 13189 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-cleq 2133 df-clel 2136 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 |
This theorem is referenced by: bdrmo 13225 |
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