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

Theorem indistopon 20093
Description: The indiscrete topology on a set  A. Part of Example 2 in [Munkres] p. 77. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
indistopon  |-  ( A  e.  V  ->  { (/) ,  A }  e.  (TopOn `  A ) )

Proof of Theorem indistopon
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sspr 4127 . . . . 5  |-  ( x 
C_  { (/) ,  A } 
<->  ( ( x  =  (/)  \/  x  =  { (/)
} )  \/  (
x  =  { A }  \/  x  =  { (/) ,  A }
) ) )
2 unieq 4198 . . . . . . . . 9  |-  ( x  =  (/)  ->  U. x  =  U. (/) )
3 uni0 4217 . . . . . . . . . 10  |-  U. (/)  =  (/)
4 0ex 4528 . . . . . . . . . . 11  |-  (/)  e.  _V
54prid1 4071 . . . . . . . . . 10  |-  (/)  e.  { (/)
,  A }
63, 5eqeltri 2545 . . . . . . . . 9  |-  U. (/)  e.  { (/)
,  A }
72, 6syl6eqel 2557 . . . . . . . 8  |-  ( x  =  (/)  ->  U. x  e.  { (/) ,  A }
)
87a1i 11 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  (/)  ->  U. x  e.  { (/) ,  A }
) )
9 unieq 4198 . . . . . . . . 9  |-  ( x  =  { (/) }  ->  U. x  =  U. { (/)
} )
104unisn 4205 . . . . . . . . . 10  |-  U. { (/)
}  =  (/)
1110, 5eqeltri 2545 . . . . . . . . 9  |-  U. { (/)
}  e.  { (/) ,  A }
129, 11syl6eqel 2557 . . . . . . . 8  |-  ( x  =  { (/) }  ->  U. x  e.  { (/) ,  A } )
1312a1i 11 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { (/) }  ->  U. x  e.  { (/)
,  A } ) )
148, 13jaod 387 . . . . . 6  |-  ( A  e.  V  ->  (
( x  =  (/)  \/  x  =  { (/) } )  ->  U. x  e.  { (/) ,  A }
) )
15 unieq 4198 . . . . . . . . . 10  |-  ( x  =  { A }  ->  U. x  =  U. { A } )
16 unisng 4206 . . . . . . . . . 10  |-  ( A  e.  V  ->  U. { A }  =  A
)
1715, 16sylan9eqr 2527 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  U. x  =  A )
18 prid2g 4070 . . . . . . . . . 10  |-  ( A  e.  V  ->  A  e.  { (/) ,  A }
)
1918adantr 472 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  A  e.  { (/) ,  A }
)
2017, 19eqeltrd 2549 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  U. x  e.  { (/) ,  A }
)
2120ex 441 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { A }  ->  U. x  e.  { (/)
,  A } ) )
22 unieq 4198 . . . . . . . . . 10  |-  ( x  =  { (/) ,  A }  ->  U. x  =  U. { (/) ,  A }
)
23 uniprg 4204 . . . . . . . . . . . 12  |-  ( (
(/)  e.  _V  /\  A  e.  V )  ->  U. { (/)
,  A }  =  ( (/)  u.  A ) )
244, 23mpan 684 . . . . . . . . . . 11  |-  ( A  e.  V  ->  U. { (/)
,  A }  =  ( (/)  u.  A ) )
25 uncom 3569 . . . . . . . . . . . 12  |-  ( (/)  u.  A )  =  ( A  u.  (/) )
26 un0 3762 . . . . . . . . . . . 12  |-  ( A  u.  (/) )  =  A
2725, 26eqtri 2493 . . . . . . . . . . 11  |-  ( (/)  u.  A )  =  A
2824, 27syl6eq 2521 . . . . . . . . . 10  |-  ( A  e.  V  ->  U. { (/)
,  A }  =  A )
2922, 28sylan9eqr 2527 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  U. x  =  A )
3018adantr 472 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  A  e.  { (/) ,  A }
)
3129, 30eqeltrd 2549 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  U. x  e.  { (/) ,  A }
)
3231ex 441 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { (/) ,  A }  ->  U. x  e.  { (/) ,  A }
) )
3321, 32jaod 387 . . . . . 6  |-  ( A  e.  V  ->  (
( x  =  { A }  \/  x  =  { (/) ,  A }
)  ->  U. x  e.  { (/) ,  A }
) )
3414, 33jaod 387 . . . . 5  |-  ( A  e.  V  ->  (
( ( x  =  (/)  \/  x  =  { (/)
} )  \/  (
x  =  { A }  \/  x  =  { (/) ,  A }
) )  ->  U. x  e.  { (/) ,  A }
) )
351, 34syl5bi 225 . . . 4  |-  ( A  e.  V  ->  (
x  C_  { (/) ,  A }  ->  U. x  e.  { (/)
,  A } ) )
3635alrimiv 1781 . . 3  |-  ( A  e.  V  ->  A. x
( x  C_  { (/) ,  A }  ->  U. x  e.  { (/) ,  A }
) )
37 vex 3034 . . . . . 6  |-  x  e. 
_V
3837elpr 3977 . . . . 5  |-  ( x  e.  { (/) ,  A } 
<->  ( x  =  (/)  \/  x  =  A ) )
39 vex 3034 . . . . . . . . 9  |-  y  e. 
_V
4039elpr 3977 . . . . . . . 8  |-  ( y  e.  { (/) ,  A } 
<->  ( y  =  (/)  \/  y  =  A ) )
41 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  y  =  (/) )
4241ineq2d 3625 . . . . . . . . . . . . 13  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  =  ( x  i^i  (/) ) )
43 in0 3763 . . . . . . . . . . . . 13  |-  ( x  i^i  (/) )  =  (/)
4442, 43syl6eq 2521 . . . . . . . . . . . 12  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  =  (/) )
4544, 5syl6eqel 2557 . . . . . . . . . . 11  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  e.  { (/) ,  A }
)
4645a1i 11 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  (/)  /\  y  =  (/) )  -> 
( x  i^i  y
)  e.  { (/) ,  A } ) )
47 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
y  =  (/) )
4847ineq2d 3625 . . . . . . . . . . . . 13  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  =  ( x  i^i  (/) ) )
4948, 43syl6eq 2521 . . . . . . . . . . . 12  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  =  (/) )
5049, 5syl6eqel 2557 . . . . . . . . . . 11  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  e.  { (/) ,  A } )
5150a1i 11 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  A  /\  y  =  (/) )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
52 simpl 464 . . . . . . . . . . . . . 14  |-  ( ( x  =  (/)  /\  y  =  A )  ->  x  =  (/) )
5352ineq1d 3624 . . . . . . . . . . . . 13  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  =  ( (/)  i^i  y
) )
54 incom 3616 . . . . . . . . . . . . . 14  |-  ( (/)  i^i  y )  =  ( y  i^i  (/) )
55 in0 3763 . . . . . . . . . . . . . 14  |-  ( y  i^i  (/) )  =  (/)
5654, 55eqtri 2493 . . . . . . . . . . . . 13  |-  ( (/)  i^i  y )  =  (/)
5753, 56syl6eq 2521 . . . . . . . . . . . 12  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  =  (/) )
5857, 5syl6eqel 2557 . . . . . . . . . . 11  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  e.  { (/) ,  A }
)
5958a1i 11 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  (/)  /\  y  =  A )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
60 ineq12 3620 . . . . . . . . . . . . . 14  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x  i^i  y
)  =  ( A  i^i  A ) )
6160adantl 473 . . . . . . . . . . . . 13  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  (
x  i^i  y )  =  ( A  i^i  A ) )
62 inidm 3632 . . . . . . . . . . . . 13  |-  ( A  i^i  A )  =  A
6361, 62syl6eq 2521 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  (
x  i^i  y )  =  A )
6418adantr 472 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  A  e.  { (/) ,  A }
)
6563, 64eqeltrd 2549 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  (
x  i^i  y )  e.  { (/) ,  A }
)
6665ex 441 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  A  /\  y  =  A )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
6746, 51, 59, 66ccased 962 . . . . . . . . 9  |-  ( A  e.  V  ->  (
( ( x  =  (/)  \/  x  =  A )  /\  ( y  =  (/)  \/  y  =  A ) )  -> 
( x  i^i  y
)  e.  { (/) ,  A } ) )
6867expdimp 444 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( x  =  (/)  \/  x  =  A ) )  -> 
( ( y  =  (/)  \/  y  =  A )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
6940, 68syl5bi 225 . . . . . . 7  |-  ( ( A  e.  V  /\  ( x  =  (/)  \/  x  =  A ) )  -> 
( y  e.  { (/)
,  A }  ->  ( x  i^i  y )  e.  { (/) ,  A } ) )
7069ralrimiv 2808 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  =  (/)  \/  x  =  A ) )  ->  A. y  e.  { (/) ,  A }  ( x  i^i  y )  e. 
{ (/) ,  A }
)
7170ex 441 . . . . 5  |-  ( A  e.  V  ->  (
( x  =  (/)  \/  x  =  A )  ->  A. y  e.  { (/)
,  A }  (
x  i^i  y )  e.  { (/) ,  A }
) )
7238, 71syl5bi 225 . . . 4  |-  ( A  e.  V  ->  (
x  e.  { (/) ,  A }  ->  A. y  e.  { (/) ,  A } 
( x  i^i  y
)  e.  { (/) ,  A } ) )
7372ralrimiv 2808 . . 3  |-  ( A  e.  V  ->  A. x  e.  { (/) ,  A } A. y  e.  { (/) ,  A }  ( x  i^i  y )  e. 
{ (/) ,  A }
)
74 prex 4642 . . . 4  |-  { (/) ,  A }  e.  _V
75 istopg 20002 . . . 4  |-  ( {
(/) ,  A }  e.  _V  ->  ( { (/)
,  A }  e.  Top 
<->  ( A. x ( x  C_  { (/) ,  A }  ->  U. x  e.  { (/)
,  A } )  /\  A. x  e. 
{ (/) ,  A } A. y  e.  { (/) ,  A }  ( x  i^i  y )  e. 
{ (/) ,  A }
) ) )
7674, 75mp1i 13 . . 3  |-  ( A  e.  V  ->  ( { (/) ,  A }  e.  Top  <->  ( A. x
( x  C_  { (/) ,  A }  ->  U. x  e.  { (/) ,  A }
)  /\  A. x  e.  { (/) ,  A } A. y  e.  { (/) ,  A }  ( x  i^i  y )  e. 
{ (/) ,  A }
) ) )
7736, 73, 76mpbir2and 936 . 2  |-  ( A  e.  V  ->  { (/) ,  A }  e.  Top )
7828eqcomd 2477 . 2  |-  ( A  e.  V  ->  A  =  U. { (/) ,  A } )
79 istopon 20017 . 2  |-  ( {
(/) ,  A }  e.  (TopOn `  A )  <->  ( { (/) ,  A }  e.  Top  /\  A  = 
U. { (/) ,  A } ) )
8077, 78, 79sylanbrc 677 1  |-  ( A  e.  V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376   A.wal 1450    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    u. cun 3388    i^i cin 3389    C_ wss 3390   (/)c0 3722   {csn 3959   {cpr 3961   U.cuni 4190   ` cfv 5589   Topctop 19994  TopOnctopon 19995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-top 19998  df-topon 20000
This theorem is referenced by:  indistop  20094  indisuni  20095  indistpsx  20102  indistpsALT  20105  indistps2ALT  20106  cnindis  20385  indishmph  20890  indistgp  21193  topdifinf  31822
  Copyright terms: Public domain W3C validator