Theorem alscn0d 49025

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 49024	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
3		n0 4358	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
4	2, 3	sylibr 234	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4  ∃wex 1775   ∈ wcel 2105   ≠ wne 2937  ∅c0 4338  ∀!walsc 49017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-ne 2938  df-dif 3965  df-nul 4339  df-alsc 49019

This theorem is referenced by: (None)