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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 13044, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 13011, 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 13006 | . . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑 |
3 | ax-bdeq 13007 | . . . . . 6 ⊢ BOUNDED 𝑥 = 𝑧 | |
4 | 1, 3 | ax-bdim 13001 | . . . . 5 ⊢ BOUNDED (𝜑 → 𝑥 = 𝑧) |
5 | 4 | ax-bdal 13005 | . . . 4 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧) |
6 | 5 | ax-bdex 13006 | . . 3 ⊢ BOUNDED ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧) |
7 | 2, 6 | ax-bdan 13002 | . 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)) |
8 | reu3 2869 | . 2 ⊢ (∃!𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧))) | |
9 | 7, 8 | bd0r 13012 | 1 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wral 2414 ∃wrex 2415 ∃!wreu 2416 BOUNDED wbd 12999 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-bd0 13000 ax-bdim 13001 ax-bdan 13002 ax-bdal 13005 ax-bdex 13006 ax-bdeq 13007 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-cleq 2130 df-clel 2133 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 |
This theorem is referenced by: bdrmo 13043 |
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