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Mirrors > Home > MPE Home > Th. List > Mathboxes > nmotru | Structured version Visualization version GIF version |
Description: There does not exist at most one set such that ⊤ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
Ref | Expression |
---|---|
nmotru | ⊢ ¬ ∃*𝑥⊤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | extru 1980 | . . 3 ⊢ ∃𝑥⊤ | |
2 | neutru 33757 | . . 3 ⊢ ¬ ∃!𝑥⊤ | |
3 | jcn 164 | . . 3 ⊢ (∃𝑥⊤ → (¬ ∃!𝑥⊤ → ¬ (∃𝑥⊤ → ∃!𝑥⊤))) | |
4 | 1, 2, 3 | mp2 9 | . 2 ⊢ ¬ (∃𝑥⊤ → ∃!𝑥⊤) |
5 | moeu 2668 | . 2 ⊢ (∃*𝑥⊤ ↔ (∃𝑥⊤ → ∃!𝑥⊤)) | |
6 | 4, 5 | mtbir 325 | 1 ⊢ ¬ ∃*𝑥⊤ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ⊤wtru 1538 ∃wex 1780 ∃*wmo 2620 ∃!weu 2653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2177 ax-nul 5212 ax-pow 5268 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-mo 2622 df-eu 2654 |
This theorem is referenced by: (None) |
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