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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 20614 | . 2 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴} | |
2 | fvex 6113 | . . . 4 ⊢ ( I ‘𝐴) ∈ V | |
3 | indistopon 20615 | . . . 4 ⊢ (( I ‘𝐴) ∈ V → {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴))) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ {∅, ( I ‘𝐴)} ∈ (TopOn‘( I ‘𝐴)) |
5 | 4 | topontopi 20546 | . 2 ⊢ {∅, ( I ‘𝐴)} ∈ Top |
6 | 1, 5 | eqeltrri 2685 | 1 ⊢ {∅, 𝐴} ∈ Top |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {cpr 4127 I cid 4948 ‘cfv 5804 Topctop 20517 TopOnctopon 20518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-top 20521 df-topon 20523 |
This theorem is referenced by: indistpsx 20624 indistps 20625 indistps2 20626 indiscld 20705 indiscon 21031 txindis 21247 indispcon 30470 onpsstopbas 31599 |
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