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Theorem en2top 18590
Description: If a topology has two elements, it is the indiscrete topology. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
en2top  |-  ( J  e.  (TopOn `  X
)  ->  ( J  ~~  2o  <->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) ) )

Proof of Theorem en2top
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  J  ~~  2o )
2 toponss 18534 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
32ad2ant2rl 748 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  x  C_  X )
4 simprl 755 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  X  =  (/) )
5 sseq0 3669 . . . . . . . . . . . . . . . . 17  |-  ( ( x  C_  X  /\  X  =  (/) )  ->  x  =  (/) )
63, 4, 5syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  x  =  (/) )
7 elsn 3891 . . . . . . . . . . . . . . . 16  |-  ( x  e.  { (/) }  <->  x  =  (/) )
86, 7sylibr 212 . . . . . . . . . . . . . . 15  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  x  e.  { (/) } )
98expr 615 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  (
x  e.  J  ->  x  e.  { (/) } ) )
109ssrdv 3362 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  C_ 
{ (/) } )
11 topontop 18531 . . . . . . . . . . . . . . . 16  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
12 0opn 18517 . . . . . . . . . . . . . . . 16  |-  ( J  e.  Top  ->  (/)  e.  J
)
1311, 12syl 16 . . . . . . . . . . . . . . 15  |-  ( J  e.  (TopOn `  X
)  ->  (/)  e.  J
)
1413ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  (/)  e.  J
)
1514snssd 4018 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  { (/) } 
C_  J )
1610, 15eqssd 3373 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  =  { (/) } )
17 0ex 4422 . . . . . . . . . . . . 13  |-  (/)  e.  _V
1817ensn1 7373 . . . . . . . . . . . 12  |-  { (/) } 
~~  1o
1916, 18syl6eqbr 4329 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  ~~  1o )
2019olcd 393 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  ( J  =  (/)  \/  J  ~~  1o ) )
21 sdom2en01 8471 . . . . . . . . . 10  |-  ( J 
~<  2o  <->  ( J  =  (/)  \/  J  ~~  1o ) )
2220, 21sylibr 212 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  ~<  2o )
23 sdomnen 7338 . . . . . . . . 9  |-  ( J 
~<  2o  ->  -.  J  ~~  2o )
2422, 23syl 16 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  -.  J  ~~  2o )
2524ex 434 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( X  =  (/)  ->  -.  J  ~~  2o ) )
2625necon2ad 2659 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( J  ~~  2o  ->  X  =/=  (/) ) )
271, 26mpd 15 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  X  =/=  (/) )
2827necomd 2695 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  (/)  =/=  X
)
2913adantr 465 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  (/)  e.  J
)
30 toponmax 18533 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
3130adantr 465 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  X  e.  J )
32 en2eqpr 8174 . . . . 5  |-  ( ( J  ~~  2o  /\  (/) 
e.  J  /\  X  e.  J )  ->  ( (/) 
=/=  X  ->  J  =  { (/) ,  X }
) )
331, 29, 31, 32syl3anc 1218 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( (/) 
=/=  X  ->  J  =  { (/) ,  X }
) )
3428, 33mpd 15 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  J  =  { (/) ,  X }
)
3534, 27jca 532 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )
36 simprl 755 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  J  =  { (/)
,  X } )
3717a1i 11 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  (/)  e.  _V )
3830adantr 465 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  X  e.  J
)
39 simprr 756 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  X  =/=  (/) )
4039necomd 2695 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  (/)  =/=  X )
41 pr2nelem 8171 . . . 4  |-  ( (
(/)  e.  _V  /\  X  e.  J  /\  (/)  =/=  X
)  ->  { (/) ,  X }  ~~  2o )
4237, 38, 40, 41syl3anc 1218 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  { (/) ,  X }  ~~  2o )
4336, 42eqbrtrd 4312 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  J  ~~  2o )
4435, 43impbida 828 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J  ~~  2o  <->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   _Vcvv 2972    C_ wss 3328   (/)c0 3637   {csn 3877   {cpr 3879   class class class wbr 4292   ` cfv 5418   1oc1o 6913   2oc2o 6914    ~~ cen 7307    ~< csdm 7309   Topctop 18498  TopOnctopon 18499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-om 6477  df-1o 6920  df-2o 6921  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-top 18503  df-topon 18506
This theorem is referenced by:  hmphindis  19370
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