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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-moae | Structured version Visualization version GIF version |
Description: Two ways to express "at most one thing exists" or, in this context equivalently, "exactly one thing exists" . The equivalence results from the presence of ax-6 1961 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 34639 and exists1 2741. Gerard Lang pointed out, that ∃𝑦∀𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (df-mo 2615, trut 1534) also means "exactly one thing exists" . (Contributed by NM, 5-Apr-2004.) State the theorem using truth constant ⊤. (Revised by BJ, 7-Oct-2022.) Reduce axiom dependencies, and use ∃*. (Revised by Wolf Lammen, 7-Mar-2023.) |
Ref | Expression |
---|---|
wl-moae | ⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-motae 34637 | . 2 ⊢ (∃*𝑥⊤ → ∀𝑥 𝑥 = 𝑦) | |
2 | hbaev 2055 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦) | |
3 | 2 | 19.8w 1974 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥 𝑥 = 𝑦) |
4 | ax-1 6 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (⊤ → 𝑥 = 𝑦)) | |
5 | 4 | alimi 1803 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥(⊤ → 𝑥 = 𝑦)) |
6 | 5 | eximi 1826 | . . . 4 ⊢ (∃𝑦∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦)) |
7 | 3, 6 | syl 17 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦)) |
8 | df-mo 2615 | . . 3 ⊢ (∃*𝑥⊤ ↔ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦)) | |
9 | 7, 8 | sylibr 235 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃*𝑥⊤) |
10 | 1, 9 | impbii 210 | 1 ⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 ⊤wtru 1529 ∃wex 1771 ∃*wmo 2613 |
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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-tru 1531 df-ex 1772 df-mo 2615 |
This theorem is referenced by: wl-euae 34639 |
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