MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rmo0 Structured version   Visualization version   GIF version

Theorem rmo0 4355
Description: Vacuous restricted at-most-one quantifier is always true. (Contributed by AV, 3-Apr-2023.)
Assertion
Ref Expression
rmo0 ∃*𝑥 ∈ ∅ 𝜑

Proof of Theorem rmo0
StepHypRef Expression
1 rex0 4353 . . 3 ¬ ∃𝑥 ∈ ∅ 𝜑
21pm2.21i 119 . 2 (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑)
3 rmo5 3395 . 2 (∃*𝑥 ∈ ∅ 𝜑 ↔ (∃𝑥 ∈ ∅ 𝜑 → ∃!𝑥 ∈ ∅ 𝜑))
42, 3mpbir 230 1 ∃*𝑥 ∈ ∅ 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wrex 3069  ∃!wreu 3373  ∃*wrmo 3374  c0 4318
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-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-dif 3947  df-nul 4319
This theorem is referenced by:  rmosn  4716
  Copyright terms: Public domain W3C validator