Theorem nulmo 2739

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 4300. (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 1947	. . 3 ⊢ Ⅎ𝑥⊥
2	1	axextmo 2738	. 2 ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥)
3		nbfal 1585	. . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ⊥))
4	3	albii 1852	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ⊥))
5	4	mobii 2575	. 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 1568  ⊥wfal 1582  ∃*wmo 2564

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566

This theorem is used by:  eu0  44345