![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nulmo | Structured version Visualization version GIF version |
Description: There exists at most one empty set. With either axnul 5060 or axnulALT 5059 or ax-nul 5061, this proves that there exists a unique empty set. In practice, once the language of classes is available, we use the stronger characterization among classes eq0 4188. (Contributed by NM, 22-Dec-2007.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.) (Proof shortened by Wolf Lammen, 26-Apr-2023.) |
Ref | Expression |
---|---|
nulmo | ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1873 | . . 3 ⊢ Ⅎ𝑥⊥ | |
2 | 1 | axextmo 2749 | . 2 ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) |
3 | nbfal 1522 | . . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ⊥)) | |
4 | 3 | albii 1782 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
5 | 4 | mobii 2559 | . 2 ⊢ (∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
6 | 2, 5 | mpbir 223 | 1 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∀wal 1505 ⊥wfal 1519 ∃*wmo 2545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |