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Theorem eu0 43510
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 4344 . . 3 ¬ 𝑥 ∈ ∅
21ax-gen 1792 . 2 𝑥 ¬ 𝑥 ∈ ∅
3 ax-nul 5312 . . 3 𝑥𝑦 ¬ 𝑦𝑥
4 nulmo 2711 . . 3 ∃*𝑥𝑦 ¬ 𝑦𝑥
5 df-eu 2567 . . 3 (∃!𝑥𝑦 ¬ 𝑦𝑥 ↔ (∃𝑥𝑦 ¬ 𝑦𝑥 ∧ ∃*𝑥𝑦 ¬ 𝑦𝑥))
63, 4, 5mpbir2an 711 . 2 ∃!𝑥𝑦 ¬ 𝑦𝑥
72, 6pm3.2i 470 1 (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1535  wex 1776  wcel 2106  ∃*wmo 2536  ∃!weu 2566  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-dif 3966  df-nul 4340
This theorem is referenced by: (None)
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