Theorem falseral0 4081

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 2917	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		19.26 1798	. . 3 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
3		con3 149	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (¬ 𝜑 → ¬ 𝑥 ∈ 𝐴))
4	3	impcom 446	. . . . . 6 ⊢ ((¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → ¬ 𝑥 ∈ 𝐴)
5	4	alimi 1739	. . . . 5 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → ∀𝑥 ¬ 𝑥 ∈ 𝐴)
6		alnex 1706	. . . . 5 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴)
7	5, 6	sylib 208	. . . 4 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → ¬ ∃𝑥 𝑥 ∈ 𝐴)
8		notnotb 304	. . . . 5 ⊢ (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
9		neq0 3930	. . . . 5 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
10	8, 9	xchbinx 324	. . . 4 ⊢ (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴)
11	7, 10	sylibr 224	. . 3 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → 𝐴 = ∅)
12	2, 11	sylbir 225	. 2 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝐴 = ∅)
13	1, 12	sylan2b 492	1 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  uvtxa01vtx0  26297