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Theorem falseral0 4220
 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 3066 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
2 19.26 1949 . . 3 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥𝐴𝜑)))
3 con3 150 . . . . . . 7 ((𝑥𝐴𝜑) → (¬ 𝜑 → ¬ 𝑥𝐴))
43impcom 394 . . . . . 6 ((¬ 𝜑 ∧ (𝑥𝐴𝜑)) → ¬ 𝑥𝐴)
54alimi 1887 . . . . 5 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → ∀𝑥 ¬ 𝑥𝐴)
6 alnex 1854 . . . . 5 (∀𝑥 ¬ 𝑥𝐴 ↔ ¬ ∃𝑥 𝑥𝐴)
75, 6sylib 208 . . . 4 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → ¬ ∃𝑥 𝑥𝐴)
8 notnotb 304 . . . . 5 (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
9 neq0 4077 . . . . 5 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
108, 9xchbinx 323 . . . 4 (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥𝐴)
117, 10sylibr 224 . . 3 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → 𝐴 = ∅)
122, 11sylbir 225 . 2 ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥𝐴𝜑)) → 𝐴 = ∅)
131, 12sylan2b 581 1 ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥𝐴 𝜑) → 𝐴 = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382  ∀wal 1629   = wceq 1631  ∃wex 1852   ∈ wcel 2145  ∀wral 3061  ∅c0 4063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-dif 3726  df-nul 4064 This theorem is referenced by:  uvtx01vtx  26525  uvtxa01vtx0OLD  26527
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