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Theorem nulmo 2709
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.)
Assertion
Ref Expression
nulmo ∃*𝑥𝑦 ¬ 𝑦𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem nulmo
StepHypRef Expression
1 nfv 1918 . . 3 𝑥
21axextmo 2708 . 2 ∃*𝑥𝑦(𝑦𝑥 ↔ ⊥)
3 nbfal 1557 . . . 4 𝑦𝑥 ↔ (𝑦𝑥 ↔ ⊥))
43albii 1822 . . 3 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 ↔ ⊥))
54mobii 2543 . 2 (∃*𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃*𝑥𝑦(𝑦𝑥 ↔ ⊥))
62, 5mpbir 230 1 ∃*𝑥𝑦 ¬ 𝑦𝑥
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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