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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 1970 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 34772 and exists1 2746. Gerard Lang pointed out, that ∃𝑦∀𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (df-mo 2622, trut 1543) 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 34770 | . 2 ⊢ (∃*𝑥⊤ → ∀𝑥 𝑥 = 𝑦) | |
2 | hbaev 2064 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦) | |
3 | 2 | 19.8w 1983 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑦∀𝑥 𝑥 = 𝑦) |
4 | ax-1 6 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (⊤ → 𝑥 = 𝑦)) | |
5 | 4 | alimi 1812 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥(⊤ → 𝑥 = 𝑦)) |
6 | 5 | eximi 1835 | . . . 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 2622 | . . 3 ⊢ (∃*𝑥⊤ ↔ ∃𝑦∀𝑥(⊤ → 𝑥 = 𝑦)) | |
9 | 7, 8 | sylibr 236 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃*𝑥⊤) |
10 | 1, 9 | impbii 211 | 1 ⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ⊤wtru 1538 ∃wex 1780 ∃*wmo 2620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1540 df-ex 1781 df-mo 2622 |
This theorem is referenced by: wl-euae 34772 |
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