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| Mirrors > Home > MPE Home > Th. List > 0ntop | Structured version Visualization version GIF version | ||
| Description: The empty set is not a topology. (Contributed by FL, 1-Jun-2008.) |
| Ref | Expression |
|---|---|
| 0ntop | ⊢ ¬ ∅ ∈ Top |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | noel 4338 | . 2 ⊢ ¬ ∅ ∈ ∅ | |
| 2 | 0opn 22910 | . 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅) | |
| 3 | 1, 2 | mto 197 | 1 ⊢ ¬ ∅ ∈ Top |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∈ wcel 2108 ∅c0 4333 Topctop 22899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-uni 4908 df-top 22900 |
| This theorem is referenced by: istps 22940 ordcmp 36448 onint1 36450 kelac1 43075 |
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