Theorem alscn0d 50297

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 50296	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
3		n0 4283	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
4	2, 3	sylibr 236	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4  ∃wex 1787   ∈ wcel 2121   ≠ wne 2936  ∅c0 4263  ∀!walsc 50289

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 1975  ax-7 2016  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-ne 2937  df-dif 3887  df-nul 4264  df-alsc 50291

This theorem is referenced by: (None)