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Theorem onsucconi 30657
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 30653 . . 3  |-  ( A  e.  On  ->  suc  A  e.  Top )
31, 2ax-mp 5 . 2  |-  suc  A  e.  Top
4 elin 3625 . . . 4  |-  ( x  e.  ( suc  A  i^i  ( Clsd `  suc  A ) )  <->  ( x  e.  suc  A  /\  x  e.  ( Clsd `  suc  A ) ) )
5 elsuci 4887 . . . . 5  |-  ( x  e.  suc  A  -> 
( x  e.  A  \/  x  =  A
) )
61onunisuci 4934 . . . . . . 7  |-  U. suc  A  =  A
76eqcomi 2415 . . . . . 6  |-  A  = 
U. suc  A
87cldopn 19716 . . . . 5  |-  ( x  e.  ( Clsd `  suc  A )  ->  ( A  \  x )  e.  suc  A )
91onsuci 6611 . . . . . . . . . 10  |-  suc  A  e.  On
109oneli 4928 . . . . . . . . 9  |-  ( ( A  \  x )  e.  suc  A  -> 
( A  \  x
)  e.  On )
11 elndif 3566 . . . . . . . . . . . 12  |-  ( (/)  e.  x  ->  -.  (/)  e.  ( A  \  x ) )
12 on0eln0 4876 . . . . . . . . . . . . . 14  |-  ( ( A  \  x )  e.  On  ->  ( (/) 
e.  ( A  \  x )  <->  ( A  \  x )  =/=  (/) ) )
1312biimprd 223 . . . . . . . . . . . . 13  |-  ( ( A  \  x )  e.  On  ->  (
( A  \  x
)  =/=  (/)  ->  (/)  e.  ( A  \  x ) ) )
1413necon1bd 2621 . . . . . . . . . . . 12  |-  ( ( A  \  x )  e.  On  ->  ( -.  (/)  e.  ( A 
\  x )  -> 
( A  \  x
)  =  (/) ) )
15 ssdif0 3827 . . . . . . . . . . . . 13  |-  ( A 
C_  x  <->  ( A  \  x )  =  (/) )
161onssneli 4930 . . . . . . . . . . . . 13  |-  ( A 
C_  x  ->  -.  x  e.  A )
1715, 16sylbir 213 . . . . . . . . . . . 12  |-  ( ( A  \  x )  =  (/)  ->  -.  x  e.  A )
1811, 14, 17syl56 32 . . . . . . . . . . 11  |-  ( ( A  \  x )  e.  On  ->  ( (/) 
e.  x  ->  -.  x  e.  A )
)
1918con2d 115 . . . . . . . . . 10  |-  ( ( A  \  x )  e.  On  ->  (
x  e.  A  ->  -.  (/)  e.  x ) )
201oneli 4928 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  x  e.  On )
21 on0eln0 4876 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  ( (/) 
e.  x  <->  x  =/=  (/) ) )
2221biimprd 223 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
x  =/=  (/)  ->  (/)  e.  x
) )
2320, 22syl 17 . . . . . . . . . . 11  |-  ( x  e.  A  ->  (
x  =/=  (/)  ->  (/)  e.  x
) )
2423necon1bd 2621 . . . . . . . . . 10  |-  ( x  e.  A  ->  ( -.  (/)  e.  x  ->  x  =  (/) ) )
2519, 24sylcom 27 . . . . . . . . 9  |-  ( ( A  \  x )  e.  On  ->  (
x  e.  A  ->  x  =  (/) ) )
2610, 25syl 17 . . . . . . . 8  |-  ( ( A  \  x )  e.  suc  A  -> 
( x  e.  A  ->  x  =  (/) ) )
2726orim1d 840 . . . . . . 7  |-  ( ( A  \  x )  e.  suc  A  -> 
( ( x  e.  A  \/  x  =  A )  ->  (
x  =  (/)  \/  x  =  A ) ) )
2827impcom 428 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  =  A
)  /\  ( A  \  x )  e.  suc  A )  ->  ( x  =  (/)  \/  x  =  A ) )
29 vex 3061 . . . . . . 7  |-  x  e. 
_V
3029elpr 3989 . . . . . 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 475 . . . 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 3445 . 2  |-  ( suc 
A  i^i  ( Clsd ` 
suc  A ) ) 
C_  { (/) ,  A }
357iscon2 20099 . 2  |-  ( suc 
A  e.  Con  <->  ( suc  A  e.  Top  /\  ( suc  A  i^i  ( Clsd `  suc  A ) ) 
C_  { (/) ,  A } ) )
363, 34, 35mpbir2an 921 1  |-  suc  A  e.  Con
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598    \ cdif 3410    i^i cin 3412    C_ wss 3413   (/)c0 3737   {cpr 3973   U.cuni 4190   Oncon0 4821   suc csuc 4823   ` cfv 5525   Topctop 19578   Clsdccld 19701   Conccon 20096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-iota 5489  df-fun 5527  df-fn 5528  df-fv 5533  df-topgen 14950  df-top 19583  df-bases 19585  df-cld 19704  df-con 20097
This theorem is referenced by:  onsuccon  30658
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