Theorem eu0 44246

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 4291	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	ax-gen 1825	. 2 ⊢ ∀𝑥 ¬ 𝑥 ∈ ∅
3		ax-nul 5269	. . 3 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
4		nulmo 2740	. . 3 ⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
5		df-eu 2597	. . 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 1568  ∃wex 1809   ∈ wcel 2143  ∃*wmo 2565  ∃!weu 2596  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-dif 3908  df-nul 4287

This theorem is referenced by: (None)