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Theorem falseral0 4114
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 2946 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
2 19.26 1838 . . 3 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥𝐴𝜑)))
3 con3 149 . . . . . . 7 ((𝑥𝐴𝜑) → (¬ 𝜑 → ¬ 𝑥𝐴))
43impcom 445 . . . . . 6 ((¬ 𝜑 ∧ (𝑥𝐴𝜑)) → ¬ 𝑥𝐴)
54alimi 1779 . . . . 5 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → ∀𝑥 ¬ 𝑥𝐴)
6 alnex 1746 . . . . 5 (∀𝑥 ¬ 𝑥𝐴 ↔ ¬ ∃𝑥 𝑥𝐴)
75, 6sylib 208 . . . 4 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → ¬ ∃𝑥 𝑥𝐴)
8 notnotb 304 . . . . 5 (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
9 neq0 3963 . . . . 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 491 1 ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥𝐴 𝜑) → 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  wral 2941  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-dif 3610  df-nul 3949
This theorem is referenced by:  uvtx01vtx  26346  uvtxa01vtx0OLD  26348
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