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Theorem indistopon 16732
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 ) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem indistopon
StepHypRef Expression
1 sspr 3778 . . . . 5  |-  ( x 
C_  { (/) ,  A } 
<->  ( ( x  =  (/)  \/  x  =  { (/)
} )  \/  (
x  =  { A }  \/  x  =  { (/) ,  A }
) ) )
2 unieq 3837 . . . . . . . . 9  |-  ( x  =  (/)  ->  U. x  =  U. (/) )
3 uni0 3855 . . . . . . . . . 10  |-  U. (/)  =  (/)
4 0ex 4151 . . . . . . . . . . 11  |-  (/)  e.  _V
54prid1 3735 . . . . . . . . . 10  |-  (/)  e.  { (/)
,  A }
63, 5eqeltri 2354 . . . . . . . . 9  |-  U. (/)  e.  { (/)
,  A }
72, 6syl6eqel 2372 . . . . . . . 8  |-  ( x  =  (/)  ->  U. x  e.  { (/) ,  A }
)
87a1i 12 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  (/)  ->  U. x  e.  { (/) ,  A }
) )
9 unieq 3837 . . . . . . . . 9  |-  ( x  =  { (/) }  ->  U. x  =  U. { (/)
} )
104unisn 3844 . . . . . . . . . 10  |-  U. { (/)
}  =  (/)
1110, 5eqeltri 2354 . . . . . . . . 9  |-  U. { (/)
}  e.  { (/) ,  A }
129, 11syl6eqel 2372 . . . . . . . 8  |-  ( x  =  { (/) }  ->  U. x  e.  { (/) ,  A } )
1312a1i 12 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { (/) }  ->  U. x  e.  { (/)
,  A } ) )
148, 13jaod 371 . . . . . 6  |-  ( A  e.  V  ->  (
( x  =  (/)  \/  x  =  { (/) } )  ->  U. x  e.  { (/) ,  A }
) )
15 unieq 3837 . . . . . . . . . 10  |-  ( x  =  { A }  ->  U. x  =  U. { A } )
16 unisng 3845 . . . . . . . . . 10  |-  ( A  e.  V  ->  U. { A }  =  A
)
1715, 16sylan9eqr 2338 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  U. x  =  A )
18 prid2g 3734 . . . . . . . . . 10  |-  ( A  e.  V  ->  A  e.  { (/) ,  A }
)
1918adantr 453 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  A  e.  { (/) ,  A }
)
2017, 19eqeltrd 2358 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  U. x  e.  { (/) ,  A }
)
2120ex 425 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { A }  ->  U. x  e.  { (/)
,  A } ) )
22 unieq 3837 . . . . . . . . . 10  |-  ( x  =  { (/) ,  A }  ->  U. x  =  U. { (/) ,  A }
)
23 uniprg 3843 . . . . . . . . . . . 12  |-  ( (
(/)  e.  _V  /\  A  e.  V )  ->  U. { (/)
,  A }  =  ( (/)  u.  A ) )
244, 23mpan 653 . . . . . . . . . . 11  |-  ( A  e.  V  ->  U. { (/)
,  A }  =  ( (/)  u.  A ) )
25 uncom 3320 . . . . . . . . . . . 12  |-  ( (/)  u.  A )  =  ( A  u.  (/) )
26 un0 3480 . . . . . . . . . . . 12  |-  ( A  u.  (/) )  =  A
2725, 26eqtri 2304 . . . . . . . . . . 11  |-  ( (/)  u.  A )  =  A
2824, 27syl6eq 2332 . . . . . . . . . 10  |-  ( A  e.  V  ->  U. { (/)
,  A }  =  A )
2922, 28sylan9eqr 2338 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  U. x  =  A )
3018adantr 453 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  A  e.  { (/) ,  A }
)
3129, 30eqeltrd 2358 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  U. x  e.  { (/) ,  A }
)
3231ex 425 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { (/) ,  A }  ->  U. x  e.  { (/) ,  A }
) )
3321, 32jaod 371 . . . . . 6  |-  ( A  e.  V  ->  (
( x  =  { A }  \/  x  =  { (/) ,  A }
)  ->  U. x  e.  { (/) ,  A }
) )
3414, 33jaod 371 . . . . 5  |-  ( A  e.  V  ->  (
( ( x  =  (/)  \/  x  =  { (/)
} )  \/  (
x  =  { A }  \/  x  =  { (/) ,  A }
) )  ->  U. x  e.  { (/) ,  A }
) )
351, 34syl5bi 210 . . . 4  |-  ( A  e.  V  ->  (
x  C_  { (/) ,  A }  ->  U. x  e.  { (/)
,  A } ) )
3635alrimiv 1618 . . 3  |-  ( A  e.  V  ->  A. x
( x  C_  { (/) ,  A }  ->  U. x  e.  { (/) ,  A }
) )
37 vex 2792 . . . . . 6  |-  x  e. 
_V
3837elpr 3659 . . . . 5  |-  ( x  e.  { (/) ,  A } 
<->  ( x  =  (/)  \/  x  =  A ) )
39 vex 2792 . . . . . . . . 9  |-  y  e. 
_V
4039elpr 3659 . . . . . . . 8  |-  ( y  e.  { (/) ,  A } 
<->  ( y  =  (/)  \/  y  =  A ) )
41 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  y  =  (/) )
4241ineq2d 3371 . . . . . . . . . . . . 13  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  =  ( x  i^i  (/) ) )
43 in0 3481 . . . . . . . . . . . . 13  |-  ( x  i^i  (/) )  =  (/)
4442, 43syl6eq 2332 . . . . . . . . . . . 12  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  =  (/) )
4544, 5syl6eqel 2372 . . . . . . . . . . 11  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  e.  { (/) ,  A }
)
4645a1i 12 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  (/)  /\  y  =  (/) )  -> 
( x  i^i  y
)  e.  { (/) ,  A } ) )
47 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
y  =  (/) )
4847ineq2d 3371 . . . . . . . . . . . . 13  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  =  ( x  i^i  (/) ) )
4948, 43syl6eq 2332 . . . . . . . . . . . 12  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  =  (/) )
5049, 5syl6eqel 2372 . . . . . . . . . . 11  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  e.  { (/) ,  A } )
5150a1i 12 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  A  /\  y  =  (/) )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
52 simpl 445 . . . . . . . . . . . . . 14  |-  ( ( x  =  (/)  /\  y  =  A )  ->  x  =  (/) )
5352ineq1d 3370 . . . . . . . . . . . . 13  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  =  ( (/)  i^i  y
) )
54 incom 3362 . . . . . . . . . . . . . 14  |-  ( (/)  i^i  y )  =  ( y  i^i  (/) )
55 in0 3481 . . . . . . . . . . . . . 14  |-  ( y  i^i  (/) )  =  (/)
5654, 55eqtri 2304 . . . . . . . . . . . . 13  |-  ( (/)  i^i  y )  =  (/)
5753, 56syl6eq 2332 . . . . . . . . . . . 12  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  =  (/) )
5857, 5syl6eqel 2372 . . . . . . . . . . 11  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  e.  { (/) ,  A }
)
5958a1i 12 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  (/)  /\  y  =  A )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
60 ineq12 3366 . . . . . . . . . . . . . 14  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x  i^i  y
)  =  ( A  i^i  A ) )
6160adantl 454 . . . . . . . . . . . . 13  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  (
x  i^i  y )  =  ( A  i^i  A ) )
62 inidm 3379 . . . . . . . . . . . . 13  |-  ( A  i^i  A )  =  A
6361, 62syl6eq 2332 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  (
x  i^i  y )  =  A )
6418adantr 453 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  A  e.  { (/) ,  A }
)
6563, 64eqeltrd 2358 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  (
x  i^i  y )  e.  { (/) ,  A }
)
6665ex 425 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  A  /\  y  =  A )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
6746, 51, 59, 66ccased 915 . . . . . . . . 9  |-  ( A  e.  V  ->  (
( ( x  =  (/)  \/  x  =  A )  /\  ( y  =  (/)  \/  y  =  A ) )  -> 
( x  i^i  y
)  e.  { (/) ,  A } ) )
6867expdimp 428 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( x  =  (/)  \/  x  =  A ) )  -> 
( ( y  =  (/)  \/  y  =  A )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
6940, 68syl5bi 210 . . . . . . 7  |-  ( ( A  e.  V  /\  ( x  =  (/)  \/  x  =  A ) )  -> 
( y  e.  { (/)
,  A }  ->  ( x  i^i  y )  e.  { (/) ,  A } ) )
7069ralrimiv 2626 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  =  (/)  \/  x  =  A ) )  ->  A. y  e.  { (/) ,  A }  ( x  i^i  y )  e. 
{ (/) ,  A }
)
7170ex 425 . . . . 5  |-  ( A  e.  V  ->  (
( x  =  (/)  \/  x  =  A )  ->  A. y  e.  { (/)
,  A }  (
x  i^i  y )  e.  { (/) ,  A }
) )
7238, 71syl5bi 210 . . . 4  |-  ( A  e.  V  ->  (
x  e.  { (/) ,  A }  ->  A. y  e.  { (/) ,  A } 
( x  i^i  y
)  e.  { (/) ,  A } ) )
7372ralrimiv 2626 . . 3  |-  ( A  e.  V  ->  A. x  e.  { (/) ,  A } A. y  e.  { (/) ,  A }  ( x  i^i  y )  e. 
{ (/) ,  A }
)
74 prex 4216 . . . 4  |-  { (/) ,  A }  e.  _V
75 istopg 16635 . . . 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 890 . 2  |-  ( A  e.  V  ->  { (/) ,  A }  e.  Top )
7828eqcomd 2289 . 2  |-  ( A  e.  V  ->  A  =  U. { (/) ,  A } )
79 istopon 16657 . 2  |-  ( {
(/) ,  A }  e.  (TopOn `  A )  <->  ( { (/) ,  A }  e.  Top  /\  A  = 
U. { (/) ,  A } ) )
8077, 78, 79sylanbrc 647 1  |-  ( A  e.  V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360   A.wal 1528    = wceq 1624    e. wcel 1685   A.wral 2544   _Vcvv 2789    u. cun 3151    i^i cin 3152    C_ wss 3153   (/)c0 3456   {csn 3641   {cpr 3642   U.cuni 3828   ` cfv 5221   Topctop 16625  TopOnctopon 16626
This theorem is referenced by:  indistop  16733  indisuni  16734  indistpsx  16741  indistpsALT  16744  indistps2ALT  16745  cnindis  17014  indishmph  17483  indistgp  17777
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-top 16630  df-topon 16633
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