Theorem alscn0d 46385

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 → ∀!𝑥 ∈ 𝐴𝜓)

Assertion

Ref	Expression
alscn0d	⊢ (𝜑 → 𝐴 ≠ ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem alscn0d

Step	Hyp	Ref	Expression
1		alscn0d.1	. . 3 ⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓)
2	1	alsc2d 46384	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
3		n0 4277	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
4	2, 3	sylibr 233	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∅c0 4253  ∀!walsc 46377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-ne 2943  df-dif 3886  df-nul 4254  df-alsc 46379

This theorem is referenced by: (None)