Theorem eu0 44137

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 4299	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	ax-gen 1822	. 2 ⊢ ∀𝑥 ¬ 𝑥 ∈ ∅
3		ax-nul 5271	. . 3 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
4		nulmo 2746	. . 3 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
5		df-eu 2603	. . 3 ⊢ (∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ∧ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥))
6	3, 4, 5	mpbir2an 723	. 2 ⊢ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
7	2, 6	pm3.2i 475	1 ⊢ (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ∧ wa 400  ∀wal 1565  ∃wex 1806   ∈ wcel 2149  ∃*wmo 2571  ∃!weu 2602  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-dif 3916  df-nul 4295

This theorem is referenced by: (None)