![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > moeu2 | Structured version Visualization version GIF version |
Description: Uniqueness is equivalent to non-existence or unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by Peter Mazsa, 19-Nov-2024.) |
Ref | Expression |
---|---|
moeu2 | ⊢ (∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moeu 2573 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | imor 852 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑)) | |
3 | 1, 2 | bitri 275 | 1 ⊢ (∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 846 ∃wex 1774 ∃*wmo 2528 ∃!weu 2558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-mo 2530 df-eu 2559 |
This theorem is referenced by: mopickr 37835 |
Copyright terms: Public domain | W3C validator |