MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  indistop Structured version   Unicode version

Theorem indistop 19673
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 19671 . 2  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
2 fvex 5858 . . . 4  |-  (  _I 
`  A )  e. 
_V
3 indistopon 19672 . . . 4  |-  ( (  _I  `  A )  e.  _V  ->  { (/) ,  (  _I  `  A
) }  e.  (TopOn `  (  _I  `  A
) ) )
42, 3ax-mp 5 . . 3  |-  { (/) ,  (  _I  `  A
) }  e.  (TopOn `  (  _I  `  A
) )
54topontopi 19602 . 2  |-  { (/) ,  (  _I  `  A
) }  e.  Top
61, 5eqeltrri 2539 1  |-  { (/) ,  A }  e.  Top
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1823   _Vcvv 3106   (/)c0 3783   {cpr 4018    _I cid 4779   ` cfv 5570   Topctop 19564  TopOnctopon 19565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-top 19569  df-topon 19572
This theorem is referenced by:  indistpsx  19681  indistps  19682  indistps2  19683  indiscld  19762  indiscon  20088  txindis  20304  indispcon  28946  onpsstopbas  30126
  Copyright terms: Public domain W3C validator