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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 21175 | . 2 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | |
2 | fvex 6446 | . . . 4 ⊢ ( I ‘𝐴) ∈ V | |
3 | indistopon 21176 | . . . 4 ⊢ (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)) |
5 | 4 | topontopi 21090 | . 2 ⊢ {∅, ( I ‘𝐴)} ∈ Top |
6 | 1, 5 | eqeltrri 2903 | 1 ⊢ {∅, 𝐴} ∈ Top |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2164 Vcvv 3414 ∅c0 4144 {cpr 4399 I cid 5249 ‘cfv 6123 Topctop 21068 TopOnctopon 21085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-top 21069 df-topon 21086 |
This theorem is referenced by: indistpsx 21185 indistps 21186 indistps2 21187 indiscld 21266 indisconn 21592 txindis 21808 indispconn 31751 onpsstopbas 32951 |
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