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Theorem indistop 17025
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.)
Assertion
Ref Expression
indistop  |-  { (/) ,  A }  e.  Top

Proof of Theorem indistop
StepHypRef Expression
1 indislem 17023 . 2  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
2 fvex 5705 . . . 4  |-  (  _I 
`  A )  e. 
_V
3 indistopon 17024 . . . 4  |-  ( (  _I  `  A )  e.  _V  ->  { (/) ,  (  _I  `  A
) }  e.  (TopOn `  (  _I  `  A
) ) )
42, 3ax-mp 8 . . 3  |-  { (/) ,  (  _I  `  A
) }  e.  (TopOn `  (  _I  `  A
) )
54topontopi 16955 . 2  |-  { (/) ,  (  _I  `  A
) }  e.  Top
61, 5eqeltrri 2479 1  |-  { (/) ,  A }  e.  Top
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   _Vcvv 2920   (/)c0 3592   {cpr 3779    _I cid 4457   ` cfv 5417   Topctop 16917  TopOnctopon 16918
This theorem is referenced by:  indistpsx  17033  indistps  17034  indistps2  17035  indiscld  17114  indiscon  17438  txindis  17623  indispcon  24878  onpsstopbas  26088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-top 16922  df-topon 16925
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