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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 1974 | . . 3 ⊢ ∃𝑥⊤ | |
| 2 | neutru 36409 | . . 3 ⊢ ¬ ∃!𝑥⊤ | |
| 3 | jcn 162 | . . 3 ⊢ (∃𝑥⊤ → (¬ ∃!𝑥⊤ → ¬ (∃𝑥⊤ → ∃!𝑥⊤))) | |
| 4 | 1, 2, 3 | mp2 9 | . 2 ⊢ ¬ (∃𝑥⊤ → ∃!𝑥⊤) | 
| 5 | moeu 2582 | . 2 ⊢ (∃*𝑥⊤ ↔ (∃𝑥⊤ → ∃!𝑥⊤)) | |
| 6 | 4, 5 | mtbir 323 | 1 ⊢ ¬ ∃*𝑥⊤ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ⊤wtru 1540 ∃wex 1778 ∃*wmo 2537 ∃!weu 2567 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-nul 5305 ax-pow 5364 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-mo 2539 df-eu 2568 | 
| This theorem is referenced by: (None) | 
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