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Theorem alscn0d 46203
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 46202 . 2 (𝜑 → ∃𝑥 𝑥𝐴)
3 n0 4277 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
42, 3sylibr 237 1 (𝜑𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1787  wcel 2112  wne 2942  c0 4253  ∀!walsc 46195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-ne 2943  df-dif 3885  df-nul 4254  df-alsc 46197
This theorem is referenced by: (None)
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