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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 9977, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 9944, 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 9939 | . . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑 |
3 | ax-bdeq 9940 | . . . . . 6 ⊢ BOUNDED 𝑥 = 𝑧 | |
4 | 1, 3 | ax-bdim 9934 | . . . . 5 ⊢ BOUNDED (𝜑 → 𝑥 = 𝑧) |
5 | 4 | ax-bdal 9938 | . . . 4 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧) |
6 | 5 | ax-bdex 9939 | . . 3 ⊢ BOUNDED ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧) |
7 | 2, 6 | ax-bdan 9935 | . 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)) |
8 | reu3 2731 | . 2 ⊢ (∃!𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧))) | |
9 | 7, 8 | bd0r 9945 | 1 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wral 2306 ∃wrex 2307 ∃!wreu 2308 BOUNDED wbd 9932 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-bd0 9933 ax-bdim 9934 ax-bdan 9935 ax-bdal 9938 ax-bdex 9939 ax-bdeq 9940 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-cleq 2033 df-clel 2036 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 |
This theorem is referenced by: bdrmo 9976 |
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