Theorem nulmo 2737

Description: There exists at most one empty set. With either axnul 5258 or axnulALT 5257 or ax-nul 5259, 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 4296. (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	⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Distinct variable group:   𝑥,𝑦

Proof of Theorem nulmo

Step	Hyp	Ref	Expression
1		nfv 1946	. . 3 ⊢ Ⅎ𝑥⊥
2	1	axextmo 2736	. 2 ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)
3		nbfal 1584	. . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ⊥))
4	3	albii 1851	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
5	4	mobii 2573	. 2 ⊢ (∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
6	2, 5	mpbir 234	1 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1567  ⊥wfal 1581  ∃*wmo 2562

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564

This theorem is used by:  eu0  44521