Theorem falseral0 4455

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

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		df-ral 3070	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		19.26 1876	. . 3 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
3		con3 153	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (¬ 𝜑 → ¬ 𝑥 ∈ 𝐴))
4	3	impcom 407	. . . . . 6 ⊢ ((¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → ¬ 𝑥 ∈ 𝐴)
5	4	alimi 1817	. . . . 5 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → ∀𝑥 ¬ 𝑥 ∈ 𝐴)
6		alnex 1787	. . . . 5 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴)
7	5, 6	sylib 217	. . . 4 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → ¬ ∃𝑥 𝑥 ∈ 𝐴)
8		notnotb 314	. . . . 5 ⊢ (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
9		neq0 4284	. . . . 5 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
10	8, 9	xchbinx 333	. . . 4 ⊢ (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴)
11	7, 10	sylibr 233	. . 3 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → 𝐴 = ∅)
12	2, 11	sylbir 234	. 2 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝐴 = ∅)
13	1, 12	sylan2b 593	1 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1785   ∈ wcel 2109  ∀wral 3065  ∅c0 4261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-ral 3070  df-dif 3894  df-nul 4262

This theorem is referenced by:  uvtx01vtx  27745