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Theorem onsucconi 24948
Description: A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
Hypothesis
Ref Expression
onsucconi.1  |-  A  e.  On
Assertion
Ref Expression
onsucconi  |-  suc  A  e.  Con

Proof of Theorem onsucconi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 onsucconi.1 . . 3  |-  A  e.  On
2 onsuctop 24944 . . 3  |-  ( A  e.  On  ->  suc  A  e.  Top )
31, 2ax-mp 8 . 2  |-  suc  A  e.  Top
4 elin 3371 . . . 4  |-  ( x  e.  ( suc  A  i^i  ( Clsd `  suc  A ) )  <->  ( x  e.  suc  A  /\  x  e.  ( Clsd `  suc  A ) ) )
5 elsuci 4474 . . . . 5  |-  ( x  e.  suc  A  -> 
( x  e.  A  \/  x  =  A
) )
61onunisuci 4522 . . . . . . 7  |-  U. suc  A  =  A
76eqcomi 2300 . . . . . 6  |-  A  = 
U. suc  A
87cldopn 16784 . . . . 5  |-  ( x  e.  ( Clsd `  suc  A )  ->  ( A  \  x )  e.  suc  A )
91onsuci 4645 . . . . . . . . . 10  |-  suc  A  e.  On
109oneli 4516 . . . . . . . . 9  |-  ( ( A  \  x )  e.  suc  A  -> 
( A  \  x
)  e.  On )
11 elndif 3313 . . . . . . . . . . . 12  |-  ( (/)  e.  x  ->  -.  (/)  e.  ( A  \  x ) )
12 on0eln0 4463 . . . . . . . . . . . . . 14  |-  ( ( A  \  x )  e.  On  ->  ( (/) 
e.  ( A  \  x )  <->  ( A  \  x )  =/=  (/) ) )
1312biimprd 214 . . . . . . . . . . . . 13  |-  ( ( A  \  x )  e.  On  ->  (
( A  \  x
)  =/=  (/)  ->  (/)  e.  ( A  \  x ) ) )
1413necon1bd 2527 . . . . . . . . . . . 12  |-  ( ( A  \  x )  e.  On  ->  ( -.  (/)  e.  ( A 
\  x )  -> 
( A  \  x
)  =  (/) ) )
15 ssdif0 3526 . . . . . . . . . . . . 13  |-  ( A 
C_  x  <->  ( A  \  x )  =  (/) )
161onssneli 4518 . . . . . . . . . . . . 13  |-  ( A 
C_  x  ->  -.  x  e.  A )
1715, 16sylbir 204 . . . . . . . . . . . 12  |-  ( ( A  \  x )  =  (/)  ->  -.  x  e.  A )
1811, 14, 17syl56 30 . . . . . . . . . . 11  |-  ( ( A  \  x )  e.  On  ->  ( (/) 
e.  x  ->  -.  x  e.  A )
)
1918con2d 107 . . . . . . . . . 10  |-  ( ( A  \  x )  e.  On  ->  (
x  e.  A  ->  -.  (/)  e.  x ) )
201oneli 4516 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  x  e.  On )
21 on0eln0 4463 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  ( (/) 
e.  x  <->  x  =/=  (/) ) )
2221biimprd 214 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
x  =/=  (/)  ->  (/)  e.  x
) )
2320, 22syl 15 . . . . . . . . . . 11  |-  ( x  e.  A  ->  (
x  =/=  (/)  ->  (/)  e.  x
) )
2423necon1bd 2527 . . . . . . . . . 10  |-  ( x  e.  A  ->  ( -.  (/)  e.  x  ->  x  =  (/) ) )
2519, 24sylcom 25 . . . . . . . . 9  |-  ( ( A  \  x )  e.  On  ->  (
x  e.  A  ->  x  =  (/) ) )
2610, 25syl 15 . . . . . . . 8  |-  ( ( A  \  x )  e.  suc  A  -> 
( x  e.  A  ->  x  =  (/) ) )
2726orim1d 812 . . . . . . 7  |-  ( ( A  \  x )  e.  suc  A  -> 
( ( x  e.  A  \/  x  =  A )  ->  (
x  =  (/)  \/  x  =  A ) ) )
2827impcom 419 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  =  A
)  /\  ( A  \  x )  e.  suc  A )  ->  ( x  =  (/)  \/  x  =  A ) )
29 vex 2804 . . . . . . 7  |-  x  e. 
_V
3029elpr 3671 . . . . . 6  |-  ( x  e.  { (/) ,  A } 
<->  ( x  =  (/)  \/  x  =  A ) )
3128, 30sylibr 203 . . . . 5  |-  ( ( ( x  e.  A  \/  x  =  A
)  /\  ( A  \  x )  e.  suc  A )  ->  x  e.  {
(/) ,  A }
)
325, 8, 31syl2an 463 . . . 4  |-  ( ( x  e.  suc  A  /\  x  e.  ( Clsd `  suc  A ) )  ->  x  e.  {
(/) ,  A }
)
334, 32sylbi 187 . . 3  |-  ( x  e.  ( suc  A  i^i  ( Clsd `  suc  A ) )  ->  x  e.  { (/) ,  A }
)
3433ssriv 3197 . 2  |-  ( suc 
A  i^i  ( Clsd ` 
suc  A ) ) 
C_  { (/) ,  A }
357iscon2 17156 . 2  |-  ( suc 
A  e.  Con  <->  ( suc  A  e.  Top  /\  ( suc  A  i^i  ( Clsd `  suc  A ) ) 
C_  { (/) ,  A } ) )
363, 34, 35mpbir2an 886 1  |-  suc  A  e.  Con
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   {cpr 3654   U.cuni 3843   Oncon0 4408   suc csuc 4410   ` cfv 5271   Topctop 16647   Clsdccld 16769   Conccon 17153
This theorem is referenced by:  onsuccon  24949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-topgen 13360  df-top 16652  df-bases 16654  df-cld 16772  df-con 17154
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