![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > indistop | Structured version Visualization version GIF version |
Description: The indiscrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 16-Jul-2006.) (Revised by Stefan Allan, 6-Nov-2008.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
indistop | ⊢ {∅, 𝐴} ∈ Top |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indislem 21133 | . 2 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | |
2 | fvex 6424 | . . . 4 ⊢ ( I ‘𝐴) ∈ V | |
3 | indistopon 21134 | . . . 4 ⊢ (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)) |
5 | 4 | topontopi 21048 | . 2 ⊢ {∅, ( I ‘𝐴)} ∈ Top |
6 | 1, 5 | eqeltrri 2875 | 1 ⊢ {∅, 𝐴} ∈ Top |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2157 Vcvv 3385 ∅c0 4115 {cpr 4370 I cid 5219 ‘cfv 6101 Topctop 21026 TopOnctopon 21043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-iota 6064 df-fun 6103 df-fv 6109 df-top 21027 df-topon 21044 |
This theorem is referenced by: indistpsx 21143 indistps 21144 indistps2 21145 indiscld 21224 indisconn 21550 txindis 21766 indispconn 31733 onpsstopbas 32937 |
Copyright terms: Public domain | W3C validator |