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Theorem eu0 41799
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 4291 . . 3 ¬ 𝑥 ∈ ∅
21ax-gen 1798 . 2 𝑥 ¬ 𝑥 ∈ ∅
3 ax-nul 5264 . . 3 𝑥𝑦 ¬ 𝑦𝑥
4 nulmo 2713 . . 3 ∃*𝑥𝑦 ¬ 𝑦𝑥
5 df-eu 2568 . . 3 (∃!𝑥𝑦 ¬ 𝑦𝑥 ↔ (∃𝑥𝑦 ¬ 𝑦𝑥 ∧ ∃*𝑥𝑦 ¬ 𝑦𝑥))
63, 4, 5mpbir2an 710 . 2 ∃!𝑥𝑦 ¬ 𝑦𝑥
72, 6pm3.2i 472 1 (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wal 1540  wex 1782  wcel 2107  ∃*wmo 2537  ∃!weu 2567  c0 4283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-dif 3914  df-nul 4284
This theorem is referenced by: (None)
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