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 → ∀!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 49313	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
3		n0 4353	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
4	2, 3	sylibr 234	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

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)