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Mirrors > Home > MPE Home > Th. List > 2eu5 | Structured version Visualization version GIF version |
Description: An alternate definition of double existential uniqueness (see 2eu4 2734). A mistake sometimes made in the literature is to use ∃!𝑥∃!𝑦 to mean "exactly one 𝑥 and exactly one 𝑦". (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining ∀𝑥∃*𝑦𝜑 as an additional condition. The correct definition apparently has never been published (∃* means "exists at most one"). (Contributed by NM, 26-Oct-2003.) Avoid ax-13 2381. (Revised by Wolf Lammen, 2-Oct-2023.) |
Ref | Expression |
---|---|
2eu5 | ⊢ ((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2eu1v 2730 | . . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))) | |
2 | 1 | pm5.32ri 576 | . 2 ⊢ ((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ∧ ∀𝑥∃*𝑦𝜑)) |
3 | eumo 2656 | . . . . 5 ⊢ (∃!𝑦∃𝑥𝜑 → ∃*𝑦∃𝑥𝜑) | |
4 | 3 | adantl 482 | . . . 4 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃*𝑦∃𝑥𝜑) |
5 | 2moexv 2705 | . . . 4 ⊢ (∃*𝑦∃𝑥𝜑 → ∀𝑥∃*𝑦𝜑) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∀𝑥∃*𝑦𝜑) |
7 | 6 | pm4.71i 560 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ∧ ∀𝑥∃*𝑦𝜑)) |
8 | 2eu4 2734 | . 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) | |
9 | 2, 7, 8 | 3bitr2i 300 | 1 ⊢ ((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 ∃wex 1771 ∃*wmo 2613 ∃!weu 2646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 |
This theorem is referenced by: 2reu5lem3 3745 |
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