Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mo0 Structured version   Visualization version   GIF version

Theorem mo0 48583
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 48581 . . . 4 {V} = ∅
21eqcomi 2742 . . 3 ∅ = {V}
3 eqeq1 2737 . . 3 (𝐴 = ∅ → (𝐴 = {V} ↔ ∅ = {V}))
42, 3mpbiri 258 . 2 (𝐴 = ∅ → 𝐴 = {V})
5 mosn 48582 . 2 (𝐴 = {V} → ∃*𝑥 𝑥𝐴)
64, 5syl 17 1 (𝐴 = ∅ → ∃*𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1535  wcel 2104  ∃*wmo 2534  Vcvv 3477  c0 4339  {csn 4630
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 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ral 3058  df-rex 3067  df-rmo 3376  df-reu 3377  df-v 3479  df-sbc 3792  df-dif 3966  df-nul 4340  df-sn 4631
This theorem is referenced by:  mosssn  48584  mo0sn  48585  f1omo  48612
  Copyright terms: Public domain W3C validator