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Theorem falseral0 4522
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 3060 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
2 19.26 1868 . . 3 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥𝐴𝜑)))
3 con3 153 . . . . . . 7 ((𝑥𝐴𝜑) → (¬ 𝜑 → ¬ 𝑥𝐴))
43impcom 407 . . . . . 6 ((¬ 𝜑 ∧ (𝑥𝐴𝜑)) → ¬ 𝑥𝐴)
54alimi 1808 . . . . 5 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → ∀𝑥 ¬ 𝑥𝐴)
6 alnex 1778 . . . . 5 (∀𝑥 ¬ 𝑥𝐴 ↔ ¬ ∃𝑥 𝑥𝐴)
75, 6sylib 218 . . . 4 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → ¬ ∃𝑥 𝑥𝐴)
8 notnotb 315 . . . . 5 (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
9 neq0 4358 . . . . 5 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
108, 9xchbinx 334 . . . 4 (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥𝐴)
117, 10sylibr 234 . . 3 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → 𝐴 = ∅)
122, 11sylbir 235 . 2 ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥𝐴𝜑)) → 𝐴 = ∅)
131, 12sylan2b 594 1 ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥𝐴 𝜑) → 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  wral 3059  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-ral 3060  df-dif 3966  df-nul 4340
This theorem is referenced by:  uvtx01vtx  29429
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