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Theorem mo0 47856
Description: "At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
Assertion
Ref Expression
mo0 (𝐴 = ∅ → ∃*𝑥 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem mo0
StepHypRef Expression
1 vsn 47854 . . . 4 {V} = ∅
21eqcomi 2736 . . 3 ∅ = {V}
3 eqeq1 2731 . . 3 (𝐴 = ∅ → (𝐴 = {V} ↔ ∅ = {V}))
42, 3mpbiri 258 . 2 (𝐴 = ∅ → 𝐴 = {V})
5 mosn 47855 . 2 (𝐴 = {V} → ∃*𝑥 𝑥𝐴)
64, 5syl 17 1 (𝐴 = ∅ → ∃*𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  ∃*wmo 2527  Vcvv 3469  c0 4318  {csn 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-v 3471  df-sbc 3775  df-dif 3947  df-nul 4319  df-sn 4625
This theorem is referenced by:  mosssn  47857  mo0sn  47858  f1omo  47885
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