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 42350
 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 42349 . 2 (𝜑 → ∃𝑥 𝑥𝐴)
3 n0 3890 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
42, 3sylibr 223 1 (𝜑𝐴 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ∀!walsc 42342 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-nul 3875  df-alsc 42344 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator