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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 22485 | . 2 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | |
2 | fvex 6901 | . . . 4 ⊢ ( I ‘𝐴) ∈ V | |
3 | indistopon 22486 | . . . 4 ⊢ (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)) |
5 | 4 | topontopi 22399 | . 2 ⊢ {∅, ( I ‘𝐴)} ∈ Top |
6 | 1, 5 | eqeltrri 2831 | 1 ⊢ {∅, 𝐴} ∈ Top |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3475 ∅c0 4321 {cpr 4629 I cid 5572 ‘cfv 6540 Topctop 22377 TopOnctopon 22394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-top 22378 df-topon 22395 |
This theorem is referenced by: indistpsx 22495 indistps 22496 indistps2 22497 indiscld 22577 indisconn 22904 txindis 23120 indispconn 34163 onpsstopbas 35253 |
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