Users' Mathboxes Mathbox for David A. Wheeler < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  alscn0d Structured version   Visualization version   GIF version

Theorem alscn0d 48098
Description: Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018.)
Hypothesis
Ref Expression
alscn0d.1 (𝜑 → ∀!𝑥𝐴𝜓)
Assertion
Ref Expression
alscn0d (𝜑𝐴 ≠ ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem alscn0d
StepHypRef Expression
1 alscn0d.1 . . 3 (𝜑 → ∀!𝑥𝐴𝜓)
21alsc2d 48097 . 2 (𝜑 → ∃𝑥 𝑥𝐴)
3 n0 4341 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
42, 3sylibr 233 1 (𝜑𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1773  wcel 2098  wne 2934  c0 4317  ∀!walsc 48090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-ne 2935  df-dif 3946  df-nul 4318  df-alsc 48092
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator