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Theorem alscn0d 49314
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 49313 . 2 (𝜑 → ∃𝑥 𝑥𝐴)
3 n0 4353 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
42, 3sylibr 234 1 (𝜑𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1779  wcel 2108  wne 2940  c0 4333  ∀!walsc 49306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-ne 2941  df-dif 3954  df-nul 4334  df-alsc 49308
This theorem is referenced by: (None)
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