Theorem ralsn0d 50455

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 50454	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
3		rexn0 4459	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → 𝐴 ≠ ∅)
4	2, 3	syl 18	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ≠ wne 2964  ∃wrex 3095  ∅c0 4294  ∀∃wrals 50445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-ne 2965  df-ral 3086  df-rex 3096  df-dif 3916  df-nul 4295  df-rals 50447

This theorem is referenced by: (None)