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Theorem nulmo 2750
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.)
Assertion
Ref Expression
nulmo ∃*𝑥𝑦 ¬ 𝑦𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem nulmo
StepHypRef Expression
1 nfv 1873 . . 3 𝑥
21axextmo 2749 . 2 ∃*𝑥𝑦(𝑦𝑥 ↔ ⊥)
3 nbfal 1522 . . . 4 𝑦𝑥 ↔ (𝑦𝑥 ↔ ⊥))
43albii 1782 . . 3 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 ↔ ⊥))
54mobii 2559 . 2 (∃*𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃*𝑥𝑦(𝑦𝑥 ↔ ⊥))
62, 5mpbir 223 1 ∃*𝑥𝑦 ¬ 𝑦𝑥
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)
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