Mathbox for Alexander van der Vekens |
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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 4859 | . . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) | |
2 | exnal 1754 | . . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) | |
3 | 1, 2 | sylib 208 | . 2 ⊢ (∃!𝑥𝜑 → ¬ ∀𝑥𝜑) |
4 | 3 | con2i 134 | 1 ⊢ (∀𝑥𝜑 → ¬ ∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 ∃wex 1704 ∃!weu 2470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-nul 4789 ax-pow 4843 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-eu 2474 df-mo 2475 |
This theorem is referenced by: eu2ndop1stv 41202 |
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