Theorem ralsn0d 51321

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.) (Revised by David A. Wheeler, 12-Jul-2026.)

Hypothesis

Ref	Expression
ralsn0d.1	⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))

Assertion

Ref	Expression
ralsn0d	⊢ (𝜑 → 𝐴 ≠ ∅)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem ralsn0d

Step	Hyp	Ref	Expression
1		ralsn0d.1	. . 3 ⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))
2	1	rals2d 51320	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
3		rexn0 4455	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → 𝐴 ≠ ∅)
4	2, 3	syl 18	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ≠ wne 2959  ∃wrex 3090  ∅c0 4281  ∀∃wrals 51311

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-ne 2960  df-ral 3081  df-rex 3091  df-dif 4050  df-nul 4282  df-rals 51313

This theorem is used by: (None)