![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > alneu | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
alneu | ⊢ (∀𝑥𝜑 → ¬ ∃!𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eunex 5089 | . . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) | |
2 | exnal 1925 | . . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) | |
3 | 1, 2 | sylib 210 | . 2 ⊢ (∃!𝑥𝜑 → ¬ ∀𝑥𝜑) |
4 | 3 | con2i 137 | 1 ⊢ (∀𝑥𝜑 → ¬ ∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1654 ∃wex 1878 ∃!weu 2639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-nul 5013 ax-pow 5065 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-mo 2605 df-eu 2640 |
This theorem is referenced by: eu2ndop1stv 42020 |
Copyright terms: Public domain | W3C validator |