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Theorem nulmo 2795
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.)
Assertion
Ref Expression
nulmo ∃*𝑥𝑦 ¬ 𝑦𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem nulmo
StepHypRef Expression
1 nfv 1906 . . 3 𝑥
21axextmo 2794 . 2 ∃*𝑥𝑦(𝑦𝑥 ↔ ⊥)
3 nbfal 1543 . . . 4 𝑦𝑥 ↔ (𝑦𝑥 ↔ ⊥))
43albii 1811 . . 3 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 ↔ ⊥))
54mobii 2624 . 2 (∃*𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃*𝑥𝑦(𝑦𝑥 ↔ ⊥))
62, 5mpbir 232 1 ∃*𝑥𝑦 ¬ 𝑦𝑥
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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