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Theorem falseral0 4238
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 1968 . . 3 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥𝐴𝜑)))
3 con3 150 . . . . . . 7 ((𝑥𝐴𝜑) → (¬ 𝜑 → ¬ 𝑥𝐴))
43impcom 396 . . . . . 6 ((¬ 𝜑 ∧ (𝑥𝐴𝜑)) → ¬ 𝑥𝐴)
54alimi 1906 . . . . 5 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → ∀𝑥 ¬ 𝑥𝐴)
6 alnex 1876 . . . . 5 (∀𝑥 ¬ 𝑥𝐴 ↔ ¬ ∃𝑥 𝑥𝐴)
75, 6sylib 209 . . . 4 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → ¬ ∃𝑥 𝑥𝐴)
8 notnotb 306 . . . . 5 (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
9 neq0 4094 . . . . 5 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
108, 9xchbinx 325 . . . 4 (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥𝐴)
117, 10sylibr 225 . . 3 (∀𝑥𝜑 ∧ (𝑥𝐴𝜑)) → 𝐴 = ∅)
122, 11sylbir 226 . 2 ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥𝐴𝜑)) → 𝐴 = ∅)
131, 12sylan2b 587 1 ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥𝐴 𝜑) → 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1650   = wceq 1652  wex 1874  wcel 2155  wral 3055  c0 4079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-v 3352  df-dif 3735  df-nul 4080
This theorem is referenced by:  uvtx01vtx  26581  uvtxa01vtx0OLD  26583
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