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Theorem en2top 17050
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 448 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  J  ~~  2o )
2 toponss 16994 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
32ad2ant2rl 730 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  x  C_  X )
4 simprl 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  X  =  (/) )
5 sseq0 3659 . . . . . . . . . . . . . . . . 17  |-  ( ( x  C_  X  /\  X  =  (/) )  ->  x  =  (/) )
63, 4, 5syl2anc 643 . . . . . . . . . . . . . . . 16  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  x  =  (/) )
7 elsn 3829 . . . . . . . . . . . . . . . 16  |-  ( x  e.  { (/) }  <->  x  =  (/) )
86, 7sylibr 204 . . . . . . . . . . . . . . 15  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  x  e.  { (/) } )
98expr 599 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  (
x  e.  J  ->  x  e.  { (/) } ) )
109ssrdv 3354 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  C_ 
{ (/) } )
11 topontop 16991 . . . . . . . . . . . . . . . 16  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
12 0opn 16977 . . . . . . . . . . . . . . . 16  |-  ( J  e.  Top  ->  (/)  e.  J
)
1311, 12syl 16 . . . . . . . . . . . . . . 15  |-  ( J  e.  (TopOn `  X
)  ->  (/)  e.  J
)
1413ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  (/)  e.  J
)
1514snssd 3943 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  { (/) } 
C_  J )
1610, 15eqssd 3365 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  =  { (/) } )
17 0ex 4339 . . . . . . . . . . . . 13  |-  (/)  e.  _V
1817ensn1 7171 . . . . . . . . . . . 12  |-  { (/) } 
~~  1o
1916, 18syl6eqbr 4249 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  ~~  1o )
2019olcd 383 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  ( J  =  (/)  \/  J  ~~  1o ) )
21 sdom2en01 8182 . . . . . . . . . 10  |-  ( J 
~<  2o  <->  ( J  =  (/)  \/  J  ~~  1o ) )
2220, 21sylibr 204 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  ~<  2o )
23 sdomnen 7136 . . . . . . . . 9  |-  ( J 
~<  2o  ->  -.  J  ~~  2o )
2422, 23syl 16 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  -.  J  ~~  2o )
2524ex 424 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( X  =  (/)  ->  -.  J  ~~  2o ) )
2625necon2ad 2652 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( J  ~~  2o  ->  X  =/=  (/) ) )
271, 26mpd 15 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  X  =/=  (/) )
2827necomd 2687 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  (/)  =/=  X
)
2913adantr 452 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  (/)  e.  J
)
30 toponmax 16993 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
3130adantr 452 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  X  e.  J )
32 en2eqpr 7891 . . . . 5  |-  ( ( J  ~~  2o  /\  (/) 
e.  J  /\  X  e.  J )  ->  ( (/) 
=/=  X  ->  J  =  { (/) ,  X }
) )
331, 29, 31, 32syl3anc 1184 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( (/) 
=/=  X  ->  J  =  { (/) ,  X }
) )
3428, 33mpd 15 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  J  =  { (/) ,  X }
)
3534, 27jca 519 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )
36 simprl 733 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  J  =  { (/)
,  X } )
3717a1i 11 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  (/)  e.  _V )
3830adantr 452 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  X  e.  J
)
39 simprr 734 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  X  =/=  (/) )
4039necomd 2687 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  (/)  =/=  X )
41 pr2nelem 7888 . . . 4  |-  ( (
(/)  e.  _V  /\  X  e.  J  /\  (/)  =/=  X
)  ->  { (/) ,  X }  ~~  2o )
4237, 38, 40, 41syl3anc 1184 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  { (/) ,  X }  ~~  2o )
4336, 42eqbrtrd 4232 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  J  ~~  2o )
4435, 43impbida 806 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J  ~~  2o  <->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    C_ wss 3320   (/)c0 3628   {csn 3814   {cpr 3815   class class class wbr 4212   ` cfv 5454   1oc1o 6717   2oc2o 6718    ~~ cen 7106    ~< csdm 7108   Topctop 16958  TopOnctopon 16959
This theorem is referenced by:  hmphindis  17829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-2o 6725  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-top 16963  df-topon 16966
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