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Theorem indistopon 16738
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 3777 . . . . 5  |-  ( x 
C_  { (/) ,  A } 
<->  ( ( x  =  (/)  \/  x  =  { (/)
} )  \/  (
x  =  { A }  \/  x  =  { (/) ,  A }
) ) )
2 unieq 3836 . . . . . . . . 9  |-  ( x  =  (/)  ->  U. x  =  U. (/) )
3 uni0 3854 . . . . . . . . . 10  |-  U. (/)  =  (/)
4 0ex 4150 . . . . . . . . . . 11  |-  (/)  e.  _V
54prid1 3734 . . . . . . . . . 10  |-  (/)  e.  { (/)
,  A }
63, 5eqeltri 2353 . . . . . . . . 9  |-  U. (/)  e.  { (/)
,  A }
72, 6syl6eqel 2371 . . . . . . . 8  |-  ( x  =  (/)  ->  U. x  e.  { (/) ,  A }
)
87a1i 10 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  (/)  ->  U. x  e.  { (/) ,  A }
) )
9 unieq 3836 . . . . . . . . 9  |-  ( x  =  { (/) }  ->  U. x  =  U. { (/)
} )
104unisn 3843 . . . . . . . . . 10  |-  U. { (/)
}  =  (/)
1110, 5eqeltri 2353 . . . . . . . . 9  |-  U. { (/)
}  e.  { (/) ,  A }
129, 11syl6eqel 2371 . . . . . . . 8  |-  ( x  =  { (/) }  ->  U. x  e.  { (/) ,  A } )
1312a1i 10 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { (/) }  ->  U. x  e.  { (/)
,  A } ) )
148, 13jaod 369 . . . . . 6  |-  ( A  e.  V  ->  (
( x  =  (/)  \/  x  =  { (/) } )  ->  U. x  e.  { (/) ,  A }
) )
15 unieq 3836 . . . . . . . . . 10  |-  ( x  =  { A }  ->  U. x  =  U. { A } )
16 unisng 3844 . . . . . . . . . 10  |-  ( A  e.  V  ->  U. { A }  =  A
)
1715, 16sylan9eqr 2337 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  U. x  =  A )
18 prid2g 3733 . . . . . . . . . 10  |-  ( A  e.  V  ->  A  e.  { (/) ,  A }
)
1918adantr 451 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  A  e.  { (/) ,  A }
)
2017, 19eqeltrd 2357 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  U. x  e.  { (/) ,  A }
)
2120ex 423 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { A }  ->  U. x  e.  { (/)
,  A } ) )
22 unieq 3836 . . . . . . . . . 10  |-  ( x  =  { (/) ,  A }  ->  U. x  =  U. { (/) ,  A }
)
23 uniprg 3842 . . . . . . . . . . . 12  |-  ( (
(/)  e.  _V  /\  A  e.  V )  ->  U. { (/)
,  A }  =  ( (/)  u.  A ) )
244, 23mpan 651 . . . . . . . . . . 11  |-  ( A  e.  V  ->  U. { (/)
,  A }  =  ( (/)  u.  A ) )
25 uncom 3319 . . . . . . . . . . . 12  |-  ( (/)  u.  A )  =  ( A  u.  (/) )
26 un0 3479 . . . . . . . . . . . 12  |-  ( A  u.  (/) )  =  A
2725, 26eqtri 2303 . . . . . . . . . . 11  |-  ( (/)  u.  A )  =  A
2824, 27syl6eq 2331 . . . . . . . . . 10  |-  ( A  e.  V  ->  U. { (/)
,  A }  =  A )
2922, 28sylan9eqr 2337 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  U. x  =  A )
3018adantr 451 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  A  e.  { (/) ,  A }
)
3129, 30eqeltrd 2357 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  U. x  e.  { (/) ,  A }
)
3231ex 423 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { (/) ,  A }  ->  U. x  e.  { (/) ,  A }
) )
3321, 32jaod 369 . . . . . 6  |-  ( A  e.  V  ->  (
( x  =  { A }  \/  x  =  { (/) ,  A }
)  ->  U. x  e.  { (/) ,  A }
) )
3414, 33jaod 369 . . . . 5  |-  ( A  e.  V  ->  (
( ( x  =  (/)  \/  x  =  { (/)
} )  \/  (
x  =  { A }  \/  x  =  { (/) ,  A }
) )  ->  U. x  e.  { (/) ,  A }
) )
351, 34syl5bi 208 . . . 4  |-  ( A  e.  V  ->  (
x  C_  { (/) ,  A }  ->  U. x  e.  { (/)
,  A } ) )
3635alrimiv 1617 . . 3  |-  ( A  e.  V  ->  A. x
( x  C_  { (/) ,  A }  ->  U. x  e.  { (/) ,  A }
) )
37 vex 2791 . . . . . 6  |-  x  e. 
_V
3837elpr 3658 . . . . 5  |-  ( x  e.  { (/) ,  A } 
<->  ( x  =  (/)  \/  x  =  A ) )
39 vex 2791 . . . . . . . . 9  |-  y  e. 
_V
4039elpr 3658 . . . . . . . 8  |-  ( y  e.  { (/) ,  A } 
<->  ( y  =  (/)  \/  y  =  A ) )
41 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  y  =  (/) )
4241ineq2d 3370 . . . . . . . . . . . . 13  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  =  ( x  i^i  (/) ) )
43 in0 3480 . . . . . . . . . . . . 13  |-  ( x  i^i  (/) )  =  (/)
4442, 43syl6eq 2331 . . . . . . . . . . . 12  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  =  (/) )
4544, 5syl6eqel 2371 . . . . . . . . . . 11  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  e.  { (/) ,  A }
)
4645a1i 10 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  (/)  /\  y  =  (/) )  -> 
( x  i^i  y
)  e.  { (/) ,  A } ) )
47 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
y  =  (/) )
4847ineq2d 3370 . . . . . . . . . . . . 13  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  =  ( x  i^i  (/) ) )
4948, 43syl6eq 2331 . . . . . . . . . . . 12  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  =  (/) )
5049, 5syl6eqel 2371 . . . . . . . . . . 11  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  e.  { (/) ,  A } )
5150a1i 10 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  A  /\  y  =  (/) )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
52 simpl 443 . . . . . . . . . . . . . 14  |-  ( ( x  =  (/)  /\  y  =  A )  ->  x  =  (/) )
5352ineq1d 3369 . . . . . . . . . . . . 13  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  =  ( (/)  i^i  y
) )
54 incom 3361 . . . . . . . . . . . . . 14  |-  ( (/)  i^i  y )  =  ( y  i^i  (/) )
55 in0 3480 . . . . . . . . . . . . . 14  |-  ( y  i^i  (/) )  =  (/)
5654, 55eqtri 2303 . . . . . . . . . . . . 13  |-  ( (/)  i^i  y )  =  (/)
5753, 56syl6eq 2331 . . . . . . . . . . . 12  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  =  (/) )
5857, 5syl6eqel 2371 . . . . . . . . . . 11  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  e.  { (/) ,  A }
)
5958a1i 10 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  (/)  /\  y  =  A )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
60 ineq12 3365 . . . . . . . . . . . . . 14  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x  i^i  y
)  =  ( A  i^i  A ) )
6160adantl 452 . . . . . . . . . . . . 13  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  (
x  i^i  y )  =  ( A  i^i  A ) )
62 inidm 3378 . . . . . . . . . . . . 13  |-  ( A  i^i  A )  =  A
6361, 62syl6eq 2331 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  (
x  i^i  y )  =  A )
6418adantr 451 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  A  e.  { (/) ,  A }
)
6563, 64eqeltrd 2357 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  (
x  i^i  y )  e.  { (/) ,  A }
)
6665ex 423 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  A  /\  y  =  A )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
6746, 51, 59, 66ccased 913 . . . . . . . . 9  |-  ( A  e.  V  ->  (
( ( x  =  (/)  \/  x  =  A )  /\  ( y  =  (/)  \/  y  =  A ) )  -> 
( x  i^i  y
)  e.  { (/) ,  A } ) )
6867expdimp 426 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( x  =  (/)  \/  x  =  A ) )  -> 
( ( y  =  (/)  \/  y  =  A )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
6940, 68syl5bi 208 . . . . . . 7  |-  ( ( A  e.  V  /\  ( x  =  (/)  \/  x  =  A ) )  -> 
( y  e.  { (/)
,  A }  ->  ( x  i^i  y )  e.  { (/) ,  A } ) )
7069ralrimiv 2625 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  =  (/)  \/  x  =  A ) )  ->  A. y  e.  { (/) ,  A }  ( x  i^i  y )  e. 
{ (/) ,  A }
)
7170ex 423 . . . . 5  |-  ( A  e.  V  ->  (
( x  =  (/)  \/  x  =  A )  ->  A. y  e.  { (/)
,  A }  (
x  i^i  y )  e.  { (/) ,  A }
) )
7238, 71syl5bi 208 . . . 4  |-  ( A  e.  V  ->  (
x  e.  { (/) ,  A }  ->  A. y  e.  { (/) ,  A } 
( x  i^i  y
)  e.  { (/) ,  A } ) )
7372ralrimiv 2625 . . 3  |-  ( A  e.  V  ->  A. x  e.  { (/) ,  A } A. y  e.  { (/) ,  A }  ( x  i^i  y )  e. 
{ (/) ,  A }
)
74 prex 4217 . . . 4  |-  { (/) ,  A }  e.  _V
75 istopg 16641 . . . 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 11 . . 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 888 . 2  |-  ( A  e.  V  ->  { (/) ,  A }  e.  Top )
7828eqcomd 2288 . 2  |-  ( A  e.  V  ->  A  =  U. { (/) ,  A } )
79 istopon 16663 . 2  |-  ( {
(/) ,  A }  e.  (TopOn `  A )  <->  ( { (/) ,  A }  e.  Top  /\  A  = 
U. { (/) ,  A } ) )
8077, 78, 79sylanbrc 645 1  |-  ( A  e.  V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   {cpr 3641   U.cuni 3827   ` cfv 5255   Topctop 16631  TopOnctopon 16632
This theorem is referenced by:  indistop  16739  indisuni  16740  indistpsx  16747  indistpsALT  16750  indistps2ALT  16751  cnindis  17020  indishmph  17489  indistgp  17783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-top 16636  df-topon 16639
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