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Theorem nulmo 2801
 Description: There exists at most one empty set. With either axnul 5195 or axnulALT 5194 or ax-nul 5196, 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 4291. (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.)
Assertion
Ref Expression
nulmo ∃*𝑥𝑦 ¬ 𝑦𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem nulmo
StepHypRef Expression
1 nfv 1916 . . 3 𝑥
21axextmo 2800 . 2 ∃*𝑥𝑦(𝑦𝑥 ↔ ⊥)
3 nbfal 1553 . . . 4 𝑦𝑥 ↔ (𝑦𝑥 ↔ ⊥))
43albii 1821 . . 3 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 ↔ ⊥))
54mobii 2632 . 2 (∃*𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃*𝑥𝑦(𝑦𝑥 ↔ ⊥))
62, 5mpbir 234 1 ∃*𝑥𝑦 ¬ 𝑦𝑥
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1536  ⊥wfal 1550  ∃*wmo 2622 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624 This theorem is referenced by:  eu0  40144
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