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Theorem indistopon 16570
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
StepHypRef Expression
1 sspr 3677 . . . . 5  |-  ( x 
C_  { (/) ,  A } 
<->  ( ( x  =  (/)  \/  x  =  { (/)
} )  \/  (
x  =  { A }  \/  x  =  { (/) ,  A }
) ) )
2 unieq 3736 . . . . . . . . 9  |-  ( x  =  (/)  ->  U. x  =  U. (/) )
3 uni0 3752 . . . . . . . . . 10  |-  U. (/)  =  (/)
4 0ex 4047 . . . . . . . . . . 11  |-  (/)  e.  _V
54prid1 3638 . . . . . . . . . 10  |-  (/)  e.  { (/)
,  A }
63, 5eqeltri 2323 . . . . . . . . 9  |-  U. (/)  e.  { (/)
,  A }
72, 6syl6eqel 2341 . . . . . . . 8  |-  ( x  =  (/)  ->  U. x  e.  { (/) ,  A }
)
87a1i 12 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  (/)  ->  U. x  e.  { (/) ,  A }
) )
9 unieq 3736 . . . . . . . . 9  |-  ( x  =  { (/) }  ->  U. x  =  U. { (/)
} )
104unisn 3743 . . . . . . . . . 10  |-  U. { (/)
}  =  (/)
1110, 5eqeltri 2323 . . . . . . . . 9  |-  U. { (/)
}  e.  { (/) ,  A }
129, 11syl6eqel 2341 . . . . . . . 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 3736 . . . . . . . . . 10  |-  ( x  =  { A }  ->  U. x  =  U. { A } )
16 unisng 3744 . . . . . . . . . 10  |-  ( A  e.  V  ->  U. { A }  =  A
)
1715, 16sylan9eqr 2307 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  U. x  =  A )
18 prid2g 3637 . . . . . . . . . 10  |-  ( A  e.  V  ->  A  e.  { (/) ,  A }
)
1918adantr 453 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  A  e.  { (/) ,  A }
)
2017, 19eqeltrd 2327 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  U. x  e.  { (/) ,  A }
)
2120ex 425 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { A }  ->  U. x  e.  { (/)
,  A } ) )
22 unieq 3736 . . . . . . . . . 10  |-  ( x  =  { (/) ,  A }  ->  U. x  =  U. { (/) ,  A }
)
23 uniprg 3742 . . . . . . . . . . . 12  |-  ( (
(/)  e.  _V  /\  A  e.  V )  ->  U. { (/)
,  A }  =  ( (/)  u.  A ) )
244, 23mpan 654 . . . . . . . . . . 11  |-  ( A  e.  V  ->  U. { (/)
,  A }  =  ( (/)  u.  A ) )
25 uncom 3229 . . . . . . . . . . . 12  |-  ( (/)  u.  A )  =  ( A  u.  (/) )
26 un0 3386 . . . . . . . . . . . 12  |-  ( A  u.  (/) )  =  A
2725, 26eqtri 2273 . . . . . . . . . . 11  |-  ( (/)  u.  A )  =  A
2824, 27syl6eq 2301 . . . . . . . . . 10  |-  ( A  e.  V  ->  U. { (/)
,  A }  =  A )
2922, 28sylan9eqr 2307 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  U. x  =  A )
3018adantr 453 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  A  e.  { (/) ,  A }
)
3129, 30eqeltrd 2327 . . . . . . . 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 2012 . . 3  |-  ( A  e.  V  ->  A. x
( x  C_  { (/) ,  A }  ->  U. x  e.  { (/) ,  A }
) )
37 vex 2730 . . . . . 6  |-  x  e. 
_V
3837elpr 3562 . . . . 5  |-  ( x  e.  { (/) ,  A } 
<->  ( x  =  (/)  \/  x  =  A ) )
39 vex 2730 . . . . . . . . 9  |-  y  e. 
_V
4039elpr 3562 . . . . . . . 8  |-  ( y  e.  { (/) ,  A } 
<->  ( y  =  (/)  \/  y  =  A ) )
41 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  y  =  (/) )
4241ineq2d 3278 . . . . . . . . . . . . 13  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  =  ( x  i^i  (/) ) )
43 in0 3387 . . . . . . . . . . . . 13  |-  ( x  i^i  (/) )  =  (/)
4442, 43syl6eq 2301 . . . . . . . . . . . 12  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  =  (/) )
4544, 5syl6eqel 2341 . . . . . . . . . . 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 3278 . . . . . . . . . . . . 13  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  =  ( x  i^i  (/) ) )
4948, 43syl6eq 2301 . . . . . . . . . . . 12  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  =  (/) )
5049, 5syl6eqel 2341 . . . . . . . . . . 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 3277 . . . . . . . . . . . . 13  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  =  ( (/)  i^i  y
) )
54 incom 3269 . . . . . . . . . . . . . 14  |-  ( (/)  i^i  y )  =  ( y  i^i  (/) )
55 in0 3387 . . . . . . . . . . . . . 14  |-  ( y  i^i  (/) )  =  (/)
5654, 55eqtri 2273 . . . . . . . . . . . . 13  |-  ( (/)  i^i  y )  =  (/)
5753, 56syl6eq 2301 . . . . . . . . . . . 12  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  =  (/) )
5857, 5syl6eqel 2341 . . . . . . . . . . 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 3273 . . . . . . . . . . . . . 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 3285 . . . . . . . . . . . . 13  |-  ( A  i^i  A )  =  A
6361, 62syl6eq 2301 . . . . . . . . . . . 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 2327 . . . . . . . . . . 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 918 . . . . . . . . 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 2587 . . . . . 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 2587 . . 3  |-  ( A  e.  V  ->  A. x  e.  { (/) ,  A } A. y  e.  { (/) ,  A }  ( x  i^i  y )  e. 
{ (/) ,  A }
)
74 prex 4111 . . . 4  |-  { (/) ,  A }  e.  _V
75 istopg 16473 . . . 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 893 . 2  |-  ( A  e.  V  ->  { (/) ,  A }  e.  Top )
7828eqcomd 2258 . 2  |-  ( A  e.  V  ->  A  =  U. { (/) ,  A } )
79 istopon 16495 . 2  |-  ( {
(/) ,  A }  e.  (TopOn `  A )  <->  ( { (/) ,  A }  e.  Top  /\  A  = 
U. { (/) ,  A } ) )
8077, 78, 79sylanbrc 648 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 1532    = wceq 1619    e. wcel 1621   A.wral 2509   _Vcvv 2727    u. cun 3076    i^i cin 3077    C_ wss 3078   (/)c0 3362   {csn 3544   {cpr 3545   U.cuni 3727   ` cfv 4592   Topctop 16463  TopOnctopon 16464
This theorem is referenced by:  indistop  16571  indisuni  16572  indistpsx  16579  indistpsALT  16582  indistps2ALT  16583  cnindis  16852  indishmph  17321  indistgp  17615
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-top 16468  df-topon 16471
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