Theorem ralsn0d 51076

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 51075	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
3		rexn0 4451	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → 𝐴 ≠ ∅)
4	2, 3	syl 18	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ≠ wne 2955  ∃wrex 3086  ∅c0 4277  ∀∃wrals 51066

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-ne 2956  df-ral 3077  df-rex 3087  df-dif 4046  df-nul 4278  df-rals 51068

This theorem is used by: (None)