Theorem eu0 43509

Description: There is only one empty set. (Contributed by RP, 1-Oct-2023.)

Assertion

Ref	Expression
eu0	⊢ (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦

Proof of Theorem eu0

Step	Hyp	Ref	Expression
1		noel 4301	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	ax-gen 1795	. 2 ⊢ ∀𝑥 ¬ 𝑥 ∈ ∅
3		ax-nul 5261	. . 3 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
4		nulmo 2706	. . 3 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
5		df-eu 2562	. . 3 ⊢ (∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥))
6	3, 4, 5	mpbir2an 711	. 2 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
7	2, 6	pm3.2i 470	1 ⊢ (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1538  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531  ∃!weu 2561  ∅c0 4296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-dif 3917  df-nul 4297

This theorem is referenced by: (None)