Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  alneu Structured version   Visualization version   GIF version

Theorem alneu 39759
Description: If a statement holds for all sets, there is not a unique set for which the statement holds. (Contributed by Alexander van der Vekens, 28-Nov-2017.)
Assertion
Ref Expression
alneu (∀𝑥𝜑 → ¬ ∃!𝑥𝜑)

Proof of Theorem alneu
StepHypRef Expression
1 eunex 4684 . . 3 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
2 exnal 1732 . . 3 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
31, 2sylib 206 . 2 (∃!𝑥𝜑 → ¬ ∀𝑥𝜑)
43con2i 132 1 (∀𝑥𝜑 → ¬ ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472  wex 1694  ∃!weu 2362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-nul 4616  ax-pow 4668
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-eu 2366  df-mo 2367
This theorem is referenced by:  eu2ndop1stv  39760
  Copyright terms: Public domain W3C validator