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Theorem falseral0 40308
 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 ∧ ∀𝑥𝐴 𝜑) → 𝐴 = ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem falseral0
StepHypRef Expression
1 df-ral 2901 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
2 19.26 1786 . . 3 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥𝐴𝜑)))
3 con3 148 . . . . . . 7 ((𝑥𝐴𝜑) → (¬ 𝜑 → ¬ 𝑥𝐴))
43impcom 445 . . . . . 6 ((¬ 𝜑 ∧ (𝑥𝐴𝜑)) → ¬ 𝑥𝐴)
54alimi 1730 . . . . 5 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → ∀𝑥 ¬ 𝑥𝐴)
6 alnex 1697 . . . . 5 (∀𝑥 ¬ 𝑥𝐴 ↔ ¬ ∃𝑥 𝑥𝐴)
75, 6sylib 207 . . . 4 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → ¬ ∃𝑥 𝑥𝐴)
8 notnotb 303 . . . . 5 (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
9 neq0 3889 . . . . 5 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
108, 9xchbinx 323 . . . 4 (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥𝐴)
117, 10sylibr 223 . . 3 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → 𝐴 = ∅)
122, 11sylbir 224 . 2 ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥𝐴𝜑)) → 𝐴 = ∅)
131, 12sylan2b 491 1 ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥𝐴 𝜑) → 𝐴 = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-dif 3543  df-nul 3875 This theorem is referenced by:  uvtxa01vtx0  40623
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