Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sn0top | Structured version Visualization version GIF version |
Description: The singleton of the empty set is a topology. (Contributed by Stefan Allan, 3-Mar-2006.) (Proof shortened by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
sn0top | ⊢ {∅} ∈ Top |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sn0topon 21600 | . 2 ⊢ {∅} ∈ (TopOn‘∅) | |
2 | 1 | topontopi 21517 | 1 ⊢ {∅} ∈ Top |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ∅c0 4290 {csn 4560 Topctop 21495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-top 21496 df-topon 21513 |
This theorem is referenced by: restsn 21772 0cmp 21996 hmph0 22397 locfinref 31100 kur14 32458 |
Copyright terms: Public domain | W3C validator |