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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 5200 or axnulALT 5199 or ax-nul 5201, 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 4305. (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 1906 | . . 3 ⊢ Ⅎ𝑥⊥ | |
2 | 1 | axextmo 2794 | . 2 ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥) |
3 | nbfal 1543 | . . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ⊥)) | |
4 | 3 | albii 1811 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
5 | 4 | mobii 2624 | . 2 ⊢ (∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)) |
6 | 2, 5 | mpbir 232 | 1 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∀wal 1526 ⊥wfal 1540 ∃*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 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 |
This theorem is referenced by: eu0 39764 |
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