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Theorem nulmo 2701
Description: There exists at most one empty set. With either axnul 5306 or axnulALT 5305 or ax-nul 5307, 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 4343. (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 1909 . . 3 𝑥
21axextmo 2700 . 2 ∃*𝑥𝑦(𝑦𝑥 ↔ ⊥)
3 nbfal 1548 . . . 4 𝑦𝑥 ↔ (𝑦𝑥 ↔ ⊥))
43albii 1813 . . 3 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 ↔ ⊥))
54mobii 2536 . 2 (∃*𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃*𝑥𝑦(𝑦𝑥 ↔ ⊥))
62, 5mpbir 230 1 ∃*𝑥𝑦 ¬ 𝑦𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1531  wfal 1545  ∃*wmo 2526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528
This theorem is referenced by:  eu0  43097
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