Users' Mathboxes Mathbox for Chen-Pang He < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  onsucconi Structured version   Unicode version

Theorem onsucconi 28422
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 28418 . . 3  |-  ( A  e.  On  ->  suc  A  e.  Top )
31, 2ax-mp 5 . 2  |-  suc  A  e.  Top
4 elin 3642 . . . 4  |-  ( x  e.  ( suc  A  i^i  ( Clsd `  suc  A ) )  <->  ( x  e.  suc  A  /\  x  e.  ( Clsd `  suc  A ) ) )
5 elsuci 4888 . . . . 5  |-  ( x  e.  suc  A  -> 
( x  e.  A  \/  x  =  A
) )
61onunisuci 4935 . . . . . . 7  |-  U. suc  A  =  A
76eqcomi 2465 . . . . . 6  |-  A  = 
U. suc  A
87cldopn 18762 . . . . 5  |-  ( x  e.  ( Clsd `  suc  A )  ->  ( A  \  x )  e.  suc  A )
91onsuci 6554 . . . . . . . . . 10  |-  suc  A  e.  On
109oneli 4929 . . . . . . . . 9  |-  ( ( A  \  x )  e.  suc  A  -> 
( A  \  x
)  e.  On )
11 elndif 3583 . . . . . . . . . . . 12  |-  ( (/)  e.  x  ->  -.  (/)  e.  ( A  \  x ) )
12 on0eln0 4877 . . . . . . . . . . . . . 14  |-  ( ( A  \  x )  e.  On  ->  ( (/) 
e.  ( A  \  x )  <->  ( A  \  x )  =/=  (/) ) )
1312biimprd 223 . . . . . . . . . . . . 13  |-  ( ( A  \  x )  e.  On  ->  (
( A  \  x
)  =/=  (/)  ->  (/)  e.  ( A  \  x ) ) )
1413necon1bd 2667 . . . . . . . . . . . 12  |-  ( ( A  \  x )  e.  On  ->  ( -.  (/)  e.  ( A 
\  x )  -> 
( A  \  x
)  =  (/) ) )
15 ssdif0 3840 . . . . . . . . . . . . 13  |-  ( A 
C_  x  <->  ( A  \  x )  =  (/) )
161onssneli 4931 . . . . . . . . . . . . 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 4929 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  x  e.  On )
21 on0eln0 4877 . . . . . . . . . . . . 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 2667 . . . . . . . . . 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 835 . . . . . . 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 3075 . . . . . . 7  |-  x  e. 
_V
3029elpr 3998 . . . . . 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 3463 . 2  |-  ( suc 
A  i^i  ( Clsd ` 
suc  A ) ) 
C_  { (/) ,  A }
357iscon2 19145 . 2  |-  ( suc 
A  e.  Con  <->  ( suc  A  e.  Top  /\  ( suc  A  i^i  ( Clsd `  suc  A ) ) 
C_  { (/) ,  A } ) )
363, 34, 35mpbir2an 911 1  |-  suc  A  e.  Con
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645    \ cdif 3428    i^i cin 3430    C_ wss 3431   (/)c0 3740   {cpr 3982   U.cuni 4194   Oncon0 4822   suc csuc 4824   ` cfv 5521   Topctop 18625   Clsdccld 18747   Conccon 19142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fn 5524  df-fv 5529  df-topgen 14496  df-top 18630  df-bases 18632  df-cld 18750  df-con 19143
This theorem is referenced by:  onsuccon  28423
  Copyright terms: Public domain W3C validator