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Mirrors > Home > MPE Home > Th. List > nulmo | Structured version Visualization version GIF version |
Description: There exists at most one empty set. With either axnul 5266 or axnulALT 5265 or ax-nul 5267, 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 4307. (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 1918 | . . 3 ⊢ Ⅎ𝑥⊥ | |
2 | 1 | axextmo 2708 | . 2 ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) |
3 | nbfal 1557 | . . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ⊥)) | |
4 | 3 | albii 1822 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
5 | 4 | mobii 2543 | . 2 ⊢ (∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
6 | 2, 5 | mpbir 230 | 1 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1540 ⊥wfal 1554 ∃*wmo 2533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 |
This theorem is referenced by: eu0 41884 |
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