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Theorem en2top 18549
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 458 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  J  ~~  2o )
2 toponss 18493 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
32ad2ant2rl 743 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  x  C_  X )
4 simprl 750 . . . . . . . . . . . . . . . . 17  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  X  =  (/) )
5 sseq0 3666 . . . . . . . . . . . . . . . . 17  |-  ( ( x  C_  X  /\  X  =  (/) )  ->  x  =  (/) )
63, 4, 5syl2anc 656 . . . . . . . . . . . . . . . 16  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  x  =  (/) )
7 elsn 3888 . . . . . . . . . . . . . . . 16  |-  ( x  e.  { (/) }  <->  x  =  (/) )
86, 7sylibr 212 . . . . . . . . . . . . . . 15  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  ( X  =  (/)  /\  x  e.  J ) )  ->  x  e.  { (/) } )
98expr 612 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  (
x  e.  J  ->  x  e.  { (/) } ) )
109ssrdv 3359 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  C_ 
{ (/) } )
11 topontop 18490 . . . . . . . . . . . . . . . 16  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
12 0opn 18476 . . . . . . . . . . . . . . . 16  |-  ( J  e.  Top  ->  (/)  e.  J
)
1311, 12syl 16 . . . . . . . . . . . . . . 15  |-  ( J  e.  (TopOn `  X
)  ->  (/)  e.  J
)
1413ad2antrr 720 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  (/)  e.  J
)
1514snssd 4015 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  { (/) } 
C_  J )
1610, 15eqssd 3370 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  =  { (/) } )
17 0ex 4419 . . . . . . . . . . . . 13  |-  (/)  e.  _V
1817ensn1 7369 . . . . . . . . . . . 12  |-  { (/) } 
~~  1o
1916, 18syl6eqbr 4326 . . . . . . . . . . 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 8467 . . . . . . . . . 10  |-  ( J 
~<  2o  <->  ( J  =  (/)  \/  J  ~~  1o ) )
2220, 21sylibr 212 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  /\  X  =  (/) )  ->  J  ~<  2o )
23 sdomnen 7334 . . . . . . . . 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 2657 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( J  ~~  2o  ->  X  =/=  (/) ) )
271, 26mpd 15 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  X  =/=  (/) )
2827necomd 2693 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  (/)  =/=  X
)
2913adantr 462 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  (/)  e.  J
)
30 toponmax 18492 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
3130adantr 462 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  X  e.  J )
32 en2eqpr 8170 . . . . 5  |-  ( ( J  ~~  2o  /\  (/) 
e.  J  /\  X  e.  J )  ->  ( (/) 
=/=  X  ->  J  =  { (/) ,  X }
) )
331, 29, 31, 32syl3anc 1213 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( (/) 
=/=  X  ->  J  =  { (/) ,  X }
) )
3428, 33mpd 15 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  J  =  { (/) ,  X }
)
3534, 27jca 529 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  J  ~~  2o )  ->  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )
36 simprl 750 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  J  =  { (/)
,  X } )
3717a1i 11 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  (/)  e.  _V )
3830adantr 462 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  X  e.  J
)
39 simprr 751 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  X  =/=  (/) )
4039necomd 2693 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  (/)  =/=  X )
41 pr2nelem 8167 . . . 4  |-  ( (
(/)  e.  _V  /\  X  e.  J  /\  (/)  =/=  X
)  ->  { (/) ,  X }  ~~  2o )
4237, 38, 40, 41syl3anc 1213 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  { (/) ,  X }  ~~  2o )
4336, 42eqbrtrd 4309 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  =  { (/) ,  X }  /\  X  =/=  (/) ) )  ->  J  ~~  2o )
4435, 43impbida 823 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 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    C_ wss 3325   (/)c0 3634   {csn 3874   {cpr 3876   class class class wbr 4289   ` cfv 5415   1oc1o 6909   2oc2o 6910    ~~ cen 7303    ~< csdm 7305   Topctop 18457  TopOnctopon 18458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-1o 6916  df-2o 6917  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-top 18462  df-topon 18465
This theorem is referenced by:  hmphindis  19329
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