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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 10648, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 10615, 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 10610 | . . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑 |
3 | ax-bdeq 10611 | . . . . . 6 ⊢ BOUNDED 𝑥 = 𝑧 | |
4 | 1, 3 | ax-bdim 10605 | . . . . 5 ⊢ BOUNDED (𝜑 → 𝑥 = 𝑧) |
5 | 4 | ax-bdal 10609 | . . . 4 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧) |
6 | 5 | ax-bdex 10610 | . . 3 ⊢ BOUNDED ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧) |
7 | 2, 6 | ax-bdan 10606 | . 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)) |
8 | reu3 2782 | . 2 ⊢ (∃!𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧))) | |
9 | 7, 8 | bd0r 10616 | 1 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wral 2348 ∃wrex 2349 ∃!wreu 2350 BOUNDED wbd 10603 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-bd0 10604 ax-bdim 10605 ax-bdan 10606 ax-bdal 10609 ax-bdex 10610 ax-bdeq 10611 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-cleq 2074 df-clel 2077 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 |
This theorem is referenced by: bdrmo 10647 |
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