Theorem falseral0 4470

Description: A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.) (Proof shortened by TM, 16-Feb-2026.)

Assertion

Ref	Expression
falseral0	⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem falseral0

Step	Hyp	Ref	Expression
1		alral 3093	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝜑)
2		pm2.21 123	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → ⊥))
3	2	ral2imi 3103	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ⊥))
4	3	imp 410	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∀𝑥 ∈ 𝐴 ⊥)
5	1, 4	sylan 589	. 2 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∀𝑥 ∈ 𝐴 ⊥)
6		fal 1576	. . 3 ⊢ ¬ ⊥
7	6	ralf0 4453	. 2 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ 𝐴 = ∅)
8	5, 7	sylib 220	1 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1560   = wceq 1562  ⊥wfal 1574  ∀wral 3078  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-ral 3079  df-dif 3909  df-nul 4288

This theorem is referenced by:  uvtx01vtx  29600