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Mirrors > Home > MPE Home > Th. List > indistps2 | Structured version Visualization version GIF version |
Description: The indiscrete topology on a set π΄ expressed as a topological space, using direct component assignments. Compare with indistps 22377. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 22379 and indistps2ALT 22381 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.) |
Ref | Expression |
---|---|
indistps2.a | β’ (BaseβπΎ) = π΄ |
indistps2.j | β’ (TopOpenβπΎ) = {β , π΄} |
Ref | Expression |
---|---|
indistps2 | β’ πΎ β TopSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indistps2.a | . 2 β’ (BaseβπΎ) = π΄ | |
2 | indistps2.j | . 2 β’ (TopOpenβπΎ) = {β , π΄} | |
3 | 0ex 5265 | . . . 4 β’ β β V | |
4 | fvex 6856 | . . . . 5 β’ (BaseβπΎ) β V | |
5 | 1, 4 | eqeltrri 2831 | . . . 4 β’ π΄ β V |
6 | 3, 5 | unipr 4884 | . . 3 β’ βͺ {β , π΄} = (β βͺ π΄) |
7 | uncom 4114 | . . 3 β’ (β βͺ π΄) = (π΄ βͺ β ) | |
8 | un0 4351 | . . 3 β’ (π΄ βͺ β ) = π΄ | |
9 | 6, 7, 8 | 3eqtrri 2766 | . 2 β’ π΄ = βͺ {β , π΄} |
10 | indistop 22368 | . 2 β’ {β , π΄} β Top | |
11 | 1, 2, 9, 10 | istpsi 22307 | 1 β’ πΎ β TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 Vcvv 3444 βͺ cun 3909 β c0 4283 {cpr 4589 βͺ cuni 4866 βcfv 6497 Basecbs 17088 TopOpenctopn 17308 TopSpctps 22297 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-top 22259 df-topon 22276 df-topsp 22298 |
This theorem is referenced by: (None) |
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