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Theorem indistps2 16765
Description: The indiscrete topology on a set  A expressed as a topological space, using direct component assignments. Compare with indistps 16764. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 16766 and indistps2ALT 16767 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.)
Hypotheses
Ref Expression
indistps2.a  |-  ( Base `  K )  =  A
indistps2.j  |-  ( TopOpen `  K )  =  { (/)
,  A }
Assertion
Ref Expression
indistps2  |-  K  e. 
TopSp

Proof of Theorem indistps2
StepHypRef Expression
1 indistps2.a . 2  |-  ( Base `  K )  =  A
2 indistps2.j . 2  |-  ( TopOpen `  K )  =  { (/)
,  A }
3 0ex 4166 . . . 4  |-  (/)  e.  _V
4 fvex 5555 . . . . 5  |-  ( Base `  K )  e.  _V
51, 4eqeltrri 2367 . . . 4  |-  A  e. 
_V
63, 5unipr 3857 . . 3  |-  U. { (/)
,  A }  =  ( (/)  u.  A )
7 uncom 3332 . . 3  |-  ( (/)  u.  A )  =  ( A  u.  (/) )
8 un0 3492 . . 3  |-  ( A  u.  (/) )  =  A
96, 7, 83eqtrri 2321 . 2  |-  A  = 
U. { (/) ,  A }
10 indistop 16755 . 2  |-  { (/) ,  A }  e.  Top
111, 2, 9, 10istpsi 16698 1  |-  K  e. 
TopSp
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163   (/)c0 3468   {cpr 3654   U.cuni 3843   ` cfv 5271   Basecbs 13164   TopOpenctopn 13342   TopSpctps 16650
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-top 16652  df-topon 16655  df-topsp 16656
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