Theorem falseral0 4445

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 3070	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝜑)
2		pm2.21 123	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → ⊥))
3	2	ral2imi 3080	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ⊥))
4	3	imp 408	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∀𝑥 ∈ 𝐴 ⊥)
5	1, 4	sylan 587	. 2 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∀𝑥 ∈ 𝐴 ⊥)
6		fal 1562	. . 3 ⊢ ¬ ⊥
7	6	ralf0 4428	. 2 ⊢ (∀𝑥 ∈ 𝐴 ⊥ ↔ 𝐴 = ∅)
8	5, 7	sylib 220	1 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397  ∀wal 1546   = wceq 1548  ⊥wfal 1560  ∀wral 3055  ∅c0 4264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-ral 3056  df-dif 3888  df-nul 4265

This theorem is referenced by:  uvtx01vtx  29488