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Theorem indistopon 20009
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 4134 . . . . 5  |-  ( x 
C_  { (/) ,  A } 
<->  ( ( x  =  (/)  \/  x  =  { (/)
} )  \/  (
x  =  { A }  \/  x  =  { (/) ,  A }
) ) )
2 unieq 4205 . . . . . . . . 9  |-  ( x  =  (/)  ->  U. x  =  U. (/) )
3 uni0 4224 . . . . . . . . . 10  |-  U. (/)  =  (/)
4 0ex 4534 . . . . . . . . . . 11  |-  (/)  e.  _V
54prid1 4079 . . . . . . . . . 10  |-  (/)  e.  { (/)
,  A }
63, 5eqeltri 2524 . . . . . . . . 9  |-  U. (/)  e.  { (/)
,  A }
72, 6syl6eqel 2536 . . . . . . . 8  |-  ( x  =  (/)  ->  U. x  e.  { (/) ,  A }
)
87a1i 11 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  (/)  ->  U. x  e.  { (/) ,  A }
) )
9 unieq 4205 . . . . . . . . 9  |-  ( x  =  { (/) }  ->  U. x  =  U. { (/)
} )
104unisn 4212 . . . . . . . . . 10  |-  U. { (/)
}  =  (/)
1110, 5eqeltri 2524 . . . . . . . . 9  |-  U. { (/)
}  e.  { (/) ,  A }
129, 11syl6eqel 2536 . . . . . . . 8  |-  ( x  =  { (/) }  ->  U. x  e.  { (/) ,  A } )
1312a1i 11 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { (/) }  ->  U. x  e.  { (/)
,  A } ) )
148, 13jaod 382 . . . . . 6  |-  ( A  e.  V  ->  (
( x  =  (/)  \/  x  =  { (/) } )  ->  U. x  e.  { (/) ,  A }
) )
15 unieq 4205 . . . . . . . . . 10  |-  ( x  =  { A }  ->  U. x  =  U. { A } )
16 unisng 4213 . . . . . . . . . 10  |-  ( A  e.  V  ->  U. { A }  =  A
)
1715, 16sylan9eqr 2506 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  U. x  =  A )
18 prid2g 4078 . . . . . . . . . 10  |-  ( A  e.  V  ->  A  e.  { (/) ,  A }
)
1918adantr 467 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  A  e.  { (/) ,  A }
)
2017, 19eqeltrd 2528 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  =  { A } )  ->  U. x  e.  { (/) ,  A }
)
2120ex 436 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { A }  ->  U. x  e.  { (/)
,  A } ) )
22 unieq 4205 . . . . . . . . . 10  |-  ( x  =  { (/) ,  A }  ->  U. x  =  U. { (/) ,  A }
)
23 uniprg 4211 . . . . . . . . . . . 12  |-  ( (
(/)  e.  _V  /\  A  e.  V )  ->  U. { (/)
,  A }  =  ( (/)  u.  A ) )
244, 23mpan 675 . . . . . . . . . . 11  |-  ( A  e.  V  ->  U. { (/)
,  A }  =  ( (/)  u.  A ) )
25 uncom 3577 . . . . . . . . . . . 12  |-  ( (/)  u.  A )  =  ( A  u.  (/) )
26 un0 3758 . . . . . . . . . . . 12  |-  ( A  u.  (/) )  =  A
2725, 26eqtri 2472 . . . . . . . . . . 11  |-  ( (/)  u.  A )  =  A
2824, 27syl6eq 2500 . . . . . . . . . 10  |-  ( A  e.  V  ->  U. { (/)
,  A }  =  A )
2922, 28sylan9eqr 2506 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  U. x  =  A )
3018adantr 467 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  A  e.  { (/) ,  A }
)
3129, 30eqeltrd 2528 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  =  { (/) ,  A } )  ->  U. x  e.  { (/) ,  A }
)
3231ex 436 . . . . . . 7  |-  ( A  e.  V  ->  (
x  =  { (/) ,  A }  ->  U. x  e.  { (/) ,  A }
) )
3321, 32jaod 382 . . . . . 6  |-  ( A  e.  V  ->  (
( x  =  { A }  \/  x  =  { (/) ,  A }
)  ->  U. x  e.  { (/) ,  A }
) )
3414, 33jaod 382 . . . . 5  |-  ( A  e.  V  ->  (
( ( x  =  (/)  \/  x  =  { (/)
} )  \/  (
x  =  { A }  \/  x  =  { (/) ,  A }
) )  ->  U. x  e.  { (/) ,  A }
) )
351, 34syl5bi 221 . . . 4  |-  ( A  e.  V  ->  (
x  C_  { (/) ,  A }  ->  U. x  e.  { (/)
,  A } ) )
3635alrimiv 1772 . . 3  |-  ( A  e.  V  ->  A. x
( x  C_  { (/) ,  A }  ->  U. x  e.  { (/) ,  A }
) )
37 vex 3047 . . . . . 6  |-  x  e. 
_V
3837elpr 3985 . . . . 5  |-  ( x  e.  { (/) ,  A } 
<->  ( x  =  (/)  \/  x  =  A ) )
39 vex 3047 . . . . . . . . 9  |-  y  e. 
_V
4039elpr 3985 . . . . . . . 8  |-  ( y  e.  { (/) ,  A } 
<->  ( y  =  (/)  \/  y  =  A ) )
41 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  y  =  (/) )
4241ineq2d 3633 . . . . . . . . . . . . 13  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  =  ( x  i^i  (/) ) )
43 in0 3759 . . . . . . . . . . . . 13  |-  ( x  i^i  (/) )  =  (/)
4442, 43syl6eq 2500 . . . . . . . . . . . 12  |-  ( ( x  =  (/)  /\  y  =  (/) )  ->  (
x  i^i  y )  =  (/) )
4544, 5syl6eqel 2536 . . . . . . . . . . 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 463 . . . . . . . . . . . . . 14  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
y  =  (/) )
4847ineq2d 3633 . . . . . . . . . . . . 13  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  =  ( x  i^i  (/) ) )
4948, 43syl6eq 2500 . . . . . . . . . . . 12  |-  ( ( x  =  A  /\  y  =  (/) )  -> 
( x  i^i  y
)  =  (/) )
5049, 5syl6eqel 2536 . . . . . . . . . . 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 459 . . . . . . . . . . . . . 14  |-  ( ( x  =  (/)  /\  y  =  A )  ->  x  =  (/) )
5352ineq1d 3632 . . . . . . . . . . . . 13  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  =  ( (/)  i^i  y
) )
54 incom 3624 . . . . . . . . . . . . . 14  |-  ( (/)  i^i  y )  =  ( y  i^i  (/) )
55 in0 3759 . . . . . . . . . . . . . 14  |-  ( y  i^i  (/) )  =  (/)
5654, 55eqtri 2472 . . . . . . . . . . . . 13  |-  ( (/)  i^i  y )  =  (/)
5753, 56syl6eq 2500 . . . . . . . . . . . 12  |-  ( ( x  =  (/)  /\  y  =  A )  ->  (
x  i^i  y )  =  (/) )
5857, 5syl6eqel 2536 . . . . . . . . . . 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 3628 . . . . . . . . . . . . . 14  |-  ( ( x  =  A  /\  y  =  A )  ->  ( x  i^i  y
)  =  ( A  i^i  A ) )
6160adantl 468 . . . . . . . . . . . . 13  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  (
x  i^i  y )  =  ( A  i^i  A ) )
62 inidm 3640 . . . . . . . . . . . . 13  |-  ( A  i^i  A )  =  A
6361, 62syl6eq 2500 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  (
x  i^i  y )  =  A )
6418adantr 467 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  A  e.  { (/) ,  A }
)
6563, 64eqeltrd 2528 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  ( x  =  A  /\  y  =  A
) )  ->  (
x  i^i  y )  e.  { (/) ,  A }
)
6665ex 436 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( x  =  A  /\  y  =  A )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
6746, 51, 59, 66ccased 957 . . . . . . . . 9  |-  ( A  e.  V  ->  (
( ( x  =  (/)  \/  x  =  A )  /\  ( y  =  (/)  \/  y  =  A ) )  -> 
( x  i^i  y
)  e.  { (/) ,  A } ) )
6867expdimp 439 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( x  =  (/)  \/  x  =  A ) )  -> 
( ( y  =  (/)  \/  y  =  A )  ->  ( x  i^i  y )  e.  { (/)
,  A } ) )
6940, 68syl5bi 221 . . . . . . 7  |-  ( ( A  e.  V  /\  ( x  =  (/)  \/  x  =  A ) )  -> 
( y  e.  { (/)
,  A }  ->  ( x  i^i  y )  e.  { (/) ,  A } ) )
7069ralrimiv 2799 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  =  (/)  \/  x  =  A ) )  ->  A. y  e.  { (/) ,  A }  ( x  i^i  y )  e. 
{ (/) ,  A }
)
7170ex 436 . . . . 5  |-  ( A  e.  V  ->  (
( x  =  (/)  \/  x  =  A )  ->  A. y  e.  { (/)
,  A }  (
x  i^i  y )  e.  { (/) ,  A }
) )
7238, 71syl5bi 221 . . . 4  |-  ( A  e.  V  ->  (
x  e.  { (/) ,  A }  ->  A. y  e.  { (/) ,  A } 
( x  i^i  y
)  e.  { (/) ,  A } ) )
7372ralrimiv 2799 . . 3  |-  ( A  e.  V  ->  A. x  e.  { (/) ,  A } A. y  e.  { (/) ,  A }  ( x  i^i  y )  e. 
{ (/) ,  A }
)
74 prex 4641 . . . 4  |-  { (/) ,  A }  e.  _V
75 istopg 19918 . . . 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 932 . 2  |-  ( A  e.  V  ->  { (/) ,  A }  e.  Top )
7828eqcomd 2456 . 2  |-  ( A  e.  V  ->  A  =  U. { (/) ,  A } )
79 istopon 19933 . 2  |-  ( {
(/) ,  A }  e.  (TopOn `  A )  <->  ( { (/) ,  A }  e.  Top  /\  A  = 
U. { (/) ,  A } ) )
8077, 78, 79sylanbrc 669 1  |-  ( A  e.  V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371   A.wal 1441    = wceq 1443    e. wcel 1886   A.wral 2736   _Vcvv 3044    u. cun 3401    i^i cin 3402    C_ wss 3403   (/)c0 3730   {csn 3967   {cpr 3969   U.cuni 4197   ` cfv 5581   Topctop 19910  TopOnctopon 19911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-top 19914  df-topon 19916
This theorem is referenced by:  indistop  20010  indisuni  20011  indistpsx  20018  indistpsALT  20021  indistps2ALT  20022  cnindis  20301  indishmph  20806  indistgp  21108  topdifinf  31745
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