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Theorem en2top 19932
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 462 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  J  ~~  2o )
2 toponss 19875 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
32ad2ant2rl 753 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  x  C_  X )
4 simprl 762 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  X  =  (/) )
5 sseq0 3800 . . . . . . . . . . . . . . . . 17  |-  ( ( x  C_  X  /\  X  =  (/) )  ->  x  =  (/) )
63, 4, 5syl2anc 665 . . . . . . . . . . . . . . . 16  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  x  =  (/) )
7 elsn 4016 . . . . . . . . . . . . . . . 16  |-  ( x  e.  { (/) }  <->  x  =  (/) )
86, 7sylibr 215 . . . . . . . . . . . . . . 15  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  x  e.  { (/) } )
98expr 618 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  (
x  e.  J  ->  x  e.  { (/) } ) )
109ssrdv 3476 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  C_ 
{ (/) } )
11 topontop 19872 . . . . . . . . . . . . . . . 16  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
12 0opn 19865 . . . . . . . . . . . . . . . 16  |-  ( J  e.  Top  ->  (/)  e.  J
)
1311, 12syl 17 . . . . . . . . . . . . . . 15  |-  ( J  e.  (TopOn `  X
)  ->  (/)  e.  J
)
1413ad2antrr 730 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  (/)  e.  J
)
1514snssd 4148 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  { (/) } 
C_  J )
1610, 15eqssd 3487 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  =  { (/) } )
17 0ex 4557 . . . . . . . . . . . . 13  |-  (/)  e.  _V
1817ensn1 7640 . . . . . . . . . . . 12  |-  { (/) } 
~~  1o
1916, 18syl6eqbr 4463 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  ~~  1o )
2019olcd 394 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  ( J  =  (/)  \/  J  ~~  1o ) )
21 sdom2en01 8730 . . . . . . . . . 10  |-  ( J 
~<  2o  <->  ( J  =  (/)  \/  J  ~~  1o ) )
2220, 21sylibr 215 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  ~<  2o )
23 sdomnen 7605 . . . . . . . . 9  |-  ( J 
~<  2o  ->  -.  J  ~~  2o )
2422, 23syl 17 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  -.  J  ~~  2o )
2524ex 435 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( X  =  (/)  ->  -.  J  ~~  2o ) )
2625necon2ad 2644 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( J  ~~  2o  ->  X  =/=  (/) ) )
271, 26mpd 15 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  X  =/=  (/) )
2827necomd 2702 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  (/)  =/=  X
)
2913adantr 466 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  (/)  e.  J
)
30 toponmax 19874 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
3130adantr 466 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  X  e.  J )
32 en2eqpr 8437 . . . . 5  |-  ( ( J  ~~  2o  /\  (/) 
e.  J  /\  X  e.  J )  ->  ( (/) 
=/=  X  ->  J  =  { (/) ,  X }
) )
331, 29, 31, 32syl3anc 1264 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( (/) 
=/=  X  ->  J  =  { (/) ,  X }
) )
3428, 33mpd 15 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  J  =  { (/) ,  X }
)
3534, 27jca 534 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )
36 simprl 762 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  J  =  { (/)
,  X } )
3717a1i 11 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  (/)  e.  _V )
3830adantr 466 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  X  e.  J
)
39 simprr 764 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  X  =/=  (/) )
4039necomd 2702 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  (/)  =/=  X )
41 pr2nelem 8434 . . . 4  |-  ( (
(/)  e.  _V  /\  X  e.  J  /\  (/)  =/=  X
)  ->  { (/) ,  X }  ~~  2o )
4237, 38, 40, 41syl3anc 1264 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  { (/) ,  X }  ~~  2o )
4336, 42eqbrtrd 4446 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  J  ~~  2o )
4435, 43impbida 840 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J  ~~  2o  <->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   _Vcvv 3087    C_ wss 3442   (/)c0 3767   {csn 4002   {cpr 4004   class class class wbr 4426   ` cfv 5601   1oc1o 7183   2oc2o 7184    ~~ cen 7574    ~< csdm 7576   Topctop 19848  TopOnctopon 19849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-1o 7190  df-2o 7191  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-top 19852  df-topon 19854
This theorem is referenced by:  hmphindis  20743
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