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Theorem onsucconi 29479
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 29475 . . 3  |-  ( A  e.  On  ->  suc  A  e.  Top )
31, 2ax-mp 5 . 2  |-  suc  A  e.  Top
4 elin 3687 . . . 4  |-  ( x  e.  ( suc  A  i^i  ( Clsd `  suc  A ) )  <->  ( x  e.  suc  A  /\  x  e.  ( Clsd `  suc  A ) ) )
5 elsuci 4944 . . . . 5  |-  ( x  e.  suc  A  -> 
( x  e.  A  \/  x  =  A
) )
61onunisuci 4991 . . . . . . 7  |-  U. suc  A  =  A
76eqcomi 2480 . . . . . 6  |-  A  = 
U. suc  A
87cldopn 19298 . . . . 5  |-  ( x  e.  ( Clsd `  suc  A )  ->  ( A  \  x )  e.  suc  A )
91onsuci 6651 . . . . . . . . . 10  |-  suc  A  e.  On
109oneli 4985 . . . . . . . . 9  |-  ( ( A  \  x )  e.  suc  A  -> 
( A  \  x
)  e.  On )
11 elndif 3628 . . . . . . . . . . . 12  |-  ( (/)  e.  x  ->  -.  (/)  e.  ( A  \  x ) )
12 on0eln0 4933 . . . . . . . . . . . . . 14  |-  ( ( A  \  x )  e.  On  ->  ( (/) 
e.  ( A  \  x )  <->  ( A  \  x )  =/=  (/) ) )
1312biimprd 223 . . . . . . . . . . . . 13  |-  ( ( A  \  x )  e.  On  ->  (
( A  \  x
)  =/=  (/)  ->  (/)  e.  ( A  \  x ) ) )
1413necon1bd 2685 . . . . . . . . . . . 12  |-  ( ( A  \  x )  e.  On  ->  ( -.  (/)  e.  ( A 
\  x )  -> 
( A  \  x
)  =  (/) ) )
15 ssdif0 3885 . . . . . . . . . . . . 13  |-  ( A 
C_  x  <->  ( A  \  x )  =  (/) )
161onssneli 4987 . . . . . . . . . . . . 13  |-  ( A 
C_  x  ->  -.  x  e.  A )
1715, 16sylbir 213 . . . . . . . . . . . 12  |-  ( ( A  \  x )  =  (/)  ->  -.  x  e.  A )
1811, 14, 17syl56 34 . . . . . . . . . . 11  |-  ( ( A  \  x )  e.  On  ->  ( (/) 
e.  x  ->  -.  x  e.  A )
)
1918con2d 115 . . . . . . . . . 10  |-  ( ( A  \  x )  e.  On  ->  (
x  e.  A  ->  -.  (/)  e.  x ) )
201oneli 4985 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  x  e.  On )
21 on0eln0 4933 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  ( (/) 
e.  x  <->  x  =/=  (/) ) )
2221biimprd 223 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
x  =/=  (/)  ->  (/)  e.  x
) )
2320, 22syl 16 . . . . . . . . . . 11  |-  ( x  e.  A  ->  (
x  =/=  (/)  ->  (/)  e.  x
) )
2423necon1bd 2685 . . . . . . . . . 10  |-  ( x  e.  A  ->  ( -.  (/)  e.  x  ->  x  =  (/) ) )
2519, 24sylcom 29 . . . . . . . . 9  |-  ( ( A  \  x )  e.  On  ->  (
x  e.  A  ->  x  =  (/) ) )
2610, 25syl 16 . . . . . . . 8  |-  ( ( A  \  x )  e.  suc  A  -> 
( x  e.  A  ->  x  =  (/) ) )
2726orim1d 837 . . . . . . 7  |-  ( ( A  \  x )  e.  suc  A  -> 
( ( x  e.  A  \/  x  =  A )  ->  (
x  =  (/)  \/  x  =  A ) ) )
2827impcom 430 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  =  A
)  /\  ( A  \  x )  e.  suc  A )  ->  ( x  =  (/)  \/  x  =  A ) )
29 vex 3116 . . . . . . 7  |-  x  e. 
_V
3029elpr 4045 . . . . . 6  |-  ( x  e.  { (/) ,  A } 
<->  ( x  =  (/)  \/  x  =  A ) )
3128, 30sylibr 212 . . . . 5  |-  ( ( ( x  e.  A  \/  x  =  A
)  /\  ( A  \  x )  e.  suc  A )  ->  x  e.  {
(/) ,  A }
)
325, 8, 31syl2an 477 . . . 4  |-  ( ( x  e.  suc  A  /\  x  e.  ( Clsd `  suc  A ) )  ->  x  e.  {
(/) ,  A }
)
334, 32sylbi 195 . . 3  |-  ( x  e.  ( suc  A  i^i  ( Clsd `  suc  A ) )  ->  x  e.  { (/) ,  A }
)
3433ssriv 3508 . 2  |-  ( suc 
A  i^i  ( Clsd ` 
suc  A ) ) 
C_  { (/) ,  A }
357iscon2 19681 . 2  |-  ( suc 
A  e.  Con  <->  ( suc  A  e.  Top  /\  ( suc  A  i^i  ( Clsd `  suc  A ) ) 
C_  { (/) ,  A } ) )
363, 34, 35mpbir2an 918 1  |-  suc  A  e.  Con
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   {cpr 4029   U.cuni 4245   Oncon0 4878   suc csuc 4880   ` cfv 5586   Topctop 19161   Clsdccld 19283   Conccon 19678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-topgen 14695  df-top 19166  df-bases 19168  df-cld 19286  df-con 19679
This theorem is referenced by:  onsuccon  29480
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