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Theorem eu0 42256
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 4329 . . 3 ¬ 𝑥 ∈ ∅
21ax-gen 1797 . 2 𝑥 ¬ 𝑥 ∈ ∅
3 ax-nul 5305 . . 3 𝑥𝑦 ¬ 𝑦𝑥
4 nulmo 2708 . . 3 ∃*𝑥𝑦 ¬ 𝑦𝑥
5 df-eu 2563 . . 3 (∃!𝑥𝑦 ¬ 𝑦𝑥 ↔ (∃𝑥𝑦 ¬ 𝑦𝑥 ∧ ∃*𝑥𝑦 ¬ 𝑦𝑥))
63, 4, 5mpbir2an 709 . 2 ∃!𝑥𝑦 ¬ 𝑦𝑥
72, 6pm3.2i 471 1 (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wal 1539  wex 1781  wcel 2106  ∃*wmo 2532  ∃!weu 2562  c0 4321
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-dif 3950  df-nul 4322
This theorem is referenced by: (None)
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