Theorem eu0 41127

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 4264	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	ax-gen 1798	. 2 ⊢ ∀𝑥 ¬ 𝑥 ∈ ∅
3		ax-nul 5230	. . 3 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
4		nulmo 2714	. . 3 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
5		df-eu 2569	. . 3 ⊢ (∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥))
6	3, 4, 5	mpbir2an 708	. 2 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
7	2, 6	pm3.2i 471	1 ⊢ (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ∧ wa 396  ∀wal 1537  ∃wex 1782   ∈ wcel 2106  ∃*wmo 2538  ∃!weu 2568  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-dif 3890  df-nul 4257

This theorem is referenced by: (None)