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Theorem eu0 40215
 Description: There is only one empty set. (Contributed by RP, 1-Oct-2023.)
Assertion
Ref Expression
eu0 (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥𝑦 ¬ 𝑦𝑥)
Distinct variable group:   𝑥,𝑦

Proof of Theorem eu0
StepHypRef Expression
1 noel 4250 . . 3 ¬ 𝑥 ∈ ∅
21ax-gen 1797 . 2 𝑥 ¬ 𝑥 ∈ ∅
3 ax-nul 5177 . . 3 𝑥𝑦 ¬ 𝑦𝑥
4 nulmo 2778 . . 3 ∃*𝑥𝑦 ¬ 𝑦𝑥
5 df-eu 2632 . . 3 (∃!𝑥𝑦 ¬ 𝑦𝑥 ↔ (∃𝑥𝑦 ¬ 𝑦𝑥 ∧ ∃*𝑥𝑦 ¬ 𝑦𝑥))
63, 4, 5mpbir2an 710 . 2 ∃!𝑥𝑦 ¬ 𝑦𝑥
72, 6pm3.2i 474 1 (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥𝑦 ¬ 𝑦𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2112  ∃*wmo 2599  ∃!weu 2631  ∅c0 4246 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-dif 3887  df-nul 4247 This theorem is referenced by: (None)
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