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Theorem onsucconi 26187
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 26183 . . 3  |-  ( A  e.  On  ->  suc  A  e.  Top )
31, 2ax-mp 8 . 2  |-  suc  A  e.  Top
4 elin 3530 . . . 4  |-  ( x  e.  ( suc  A  i^i  ( Clsd `  suc  A ) )  <->  ( x  e.  suc  A  /\  x  e.  ( Clsd `  suc  A ) ) )
5 elsuci 4647 . . . . 5  |-  ( x  e.  suc  A  -> 
( x  e.  A  \/  x  =  A
) )
61onunisuci 4695 . . . . . . 7  |-  U. suc  A  =  A
76eqcomi 2440 . . . . . 6  |-  A  = 
U. suc  A
87cldopn 17095 . . . . 5  |-  ( x  e.  ( Clsd `  suc  A )  ->  ( A  \  x )  e.  suc  A )
91onsuci 4818 . . . . . . . . . 10  |-  suc  A  e.  On
109oneli 4689 . . . . . . . . 9  |-  ( ( A  \  x )  e.  suc  A  -> 
( A  \  x
)  e.  On )
11 elndif 3471 . . . . . . . . . . . 12  |-  ( (/)  e.  x  ->  -.  (/)  e.  ( A  \  x ) )
12 on0eln0 4636 . . . . . . . . . . . . . 14  |-  ( ( A  \  x )  e.  On  ->  ( (/) 
e.  ( A  \  x )  <->  ( A  \  x )  =/=  (/) ) )
1312biimprd 215 . . . . . . . . . . . . 13  |-  ( ( A  \  x )  e.  On  ->  (
( A  \  x
)  =/=  (/)  ->  (/)  e.  ( A  \  x ) ) )
1413necon1bd 2672 . . . . . . . . . . . 12  |-  ( ( A  \  x )  e.  On  ->  ( -.  (/)  e.  ( A 
\  x )  -> 
( A  \  x
)  =  (/) ) )
15 ssdif0 3686 . . . . . . . . . . . . 13  |-  ( A 
C_  x  <->  ( A  \  x )  =  (/) )
161onssneli 4691 . . . . . . . . . . . . 13  |-  ( A 
C_  x  ->  -.  x  e.  A )
1715, 16sylbir 205 . . . . . . . . . . . 12  |-  ( ( A  \  x )  =  (/)  ->  -.  x  e.  A )
1811, 14, 17syl56 32 . . . . . . . . . . 11  |-  ( ( A  \  x )  e.  On  ->  ( (/) 
e.  x  ->  -.  x  e.  A )
)
1918con2d 109 . . . . . . . . . 10  |-  ( ( A  \  x )  e.  On  ->  (
x  e.  A  ->  -.  (/)  e.  x ) )
201oneli 4689 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  x  e.  On )
21 on0eln0 4636 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  ( (/) 
e.  x  <->  x  =/=  (/) ) )
2221biimprd 215 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
x  =/=  (/)  ->  (/)  e.  x
) )
2320, 22syl 16 . . . . . . . . . . 11  |-  ( x  e.  A  ->  (
x  =/=  (/)  ->  (/)  e.  x
) )
2423necon1bd 2672 . . . . . . . . . 10  |-  ( x  e.  A  ->  ( -.  (/)  e.  x  ->  x  =  (/) ) )
2519, 24sylcom 27 . . . . . . . . 9  |-  ( ( A  \  x )  e.  On  ->  (
x  e.  A  ->  x  =  (/) ) )
2610, 25syl 16 . . . . . . . 8  |-  ( ( A  \  x )  e.  suc  A  -> 
( x  e.  A  ->  x  =  (/) ) )
2726orim1d 813 . . . . . . 7  |-  ( ( A  \  x )  e.  suc  A  -> 
( ( x  e.  A  \/  x  =  A )  ->  (
x  =  (/)  \/  x  =  A ) ) )
2827impcom 420 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  =  A
)  /\  ( A  \  x )  e.  suc  A )  ->  ( x  =  (/)  \/  x  =  A ) )
29 vex 2959 . . . . . . 7  |-  x  e. 
_V
3029elpr 3832 . . . . . 6  |-  ( x  e.  { (/) ,  A } 
<->  ( x  =  (/)  \/  x  =  A ) )
3128, 30sylibr 204 . . . . 5  |-  ( ( ( x  e.  A  \/  x  =  A
)  /\  ( A  \  x )  e.  suc  A )  ->  x  e.  {
(/) ,  A }
)
325, 8, 31syl2an 464 . . . 4  |-  ( ( x  e.  suc  A  /\  x  e.  ( Clsd `  suc  A ) )  ->  x  e.  {
(/) ,  A }
)
334, 32sylbi 188 . . 3  |-  ( x  e.  ( suc  A  i^i  ( Clsd `  suc  A ) )  ->  x  e.  { (/) ,  A }
)
3433ssriv 3352 . 2  |-  ( suc 
A  i^i  ( Clsd ` 
suc  A ) ) 
C_  { (/) ,  A }
357iscon2 17477 . 2  |-  ( suc 
A  e.  Con  <->  ( suc  A  e.  Top  /\  ( suc  A  i^i  ( Clsd `  suc  A ) ) 
C_  { (/) ,  A } ) )
363, 34, 35mpbir2an 887 1  |-  suc  A  e.  Con
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   {cpr 3815   U.cuni 4015   Oncon0 4581   suc csuc 4583   ` cfv 5454   Topctop 16958   Clsdccld 17080   Conccon 17474
This theorem is referenced by:  onsuccon  26188
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-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-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-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-topgen 13667  df-top 16963  df-bases 16965  df-cld 17083  df-con 17475
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