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Theorem eu0 43533
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 4338 . . 3 ¬ 𝑥 ∈ ∅
21ax-gen 1795 . 2 𝑥 ¬ 𝑥 ∈ ∅
3 ax-nul 5306 . . 3 𝑥𝑦 ¬ 𝑦𝑥
4 nulmo 2713 . . 3 ∃*𝑥𝑦 ¬ 𝑦𝑥
5 df-eu 2569 . . 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 1538  wex 1779  wcel 2108  ∃*wmo 2538  ∃!weu 2568  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-dif 3954  df-nul 4334
This theorem is referenced by: (None)
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