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Theorem connsub 20367
Description: Two equivalent ways of saying that a subspace topology is connected. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
Assertion
Ref Expression
connsub  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  (
( Jt  S )  e.  Con  <->  A. x  e.  J  A. y  e.  J  (
( ( x  i^i 
S )  =/=  (/)  /\  (
y  i^i  S )  =/=  (/)  /\  ( x  i^i  y )  C_  ( X  \  S ) )  ->  -.  S  C_  ( x  u.  y
) ) ) )
Distinct variable groups:    x, y, J    x, S, y    x, X, y

Proof of Theorem connsub
StepHypRef Expression
1 consuba 20366 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  (
( Jt  S )  e.  Con  <->  A. x  e.  J  A. y  e.  J  (
( ( x  i^i 
S )  =/=  (/)  /\  (
y  i^i  S )  =/=  (/)  /\  ( ( x  i^i  y )  i^i  S )  =  (/) )  ->  ( ( x  u.  y )  i^i  S )  =/= 
S ) ) )
2 inss1 3688 . . . . . . 7  |-  ( x  i^i  y )  C_  x
3 toponss 19875 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
43ad2ant2r 751 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  (
x  e.  J  /\  y  e.  J )
)  ->  x  C_  X
)
52, 4syl5ss 3481 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  (
x  e.  J  /\  y  e.  J )
)  ->  ( x  i^i  y )  C_  X
)
6 reldisj 3842 . . . . . 6  |-  ( ( x  i^i  y ) 
C_  X  ->  (
( ( x  i^i  y )  i^i  S
)  =  (/)  <->  ( x  i^i  y )  C_  ( X  \  S ) ) )
75, 6syl 17 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  (
x  e.  J  /\  y  e.  J )
)  ->  ( (
( x  i^i  y
)  i^i  S )  =  (/)  <->  ( x  i^i  y )  C_  ( X  \  S ) ) )
873anbi3d 1341 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  (
x  e.  J  /\  y  e.  J )
)  ->  ( (
( x  i^i  S
)  =/=  (/)  /\  (
y  i^i  S )  =/=  (/)  /\  ( ( x  i^i  y )  i^i  S )  =  (/) )  <->  ( ( x  i^i  S )  =/=  (/)  /\  ( y  i^i 
S )  =/=  (/)  /\  (
x  i^i  y )  C_  ( X  \  S
) ) ) )
9 dfss1 3673 . . . . . . 7  |-  ( S 
C_  ( x  u.  y )  <->  ( (
x  u.  y )  i^i  S )  =  S )
109a1i 11 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  (
x  e.  J  /\  y  e.  J )
)  ->  ( S  C_  ( x  u.  y
)  <->  ( ( x  u.  y )  i^i 
S )  =  S ) )
1110bicomd 204 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  (
x  e.  J  /\  y  e.  J )
)  ->  ( (
( x  u.  y
)  i^i  S )  =  S  <->  S  C_  ( x  u.  y ) ) )
1211necon3abid 2677 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  (
x  e.  J  /\  y  e.  J )
)  ->  ( (
( x  u.  y
)  i^i  S )  =/=  S  <->  -.  S  C_  (
x  u.  y ) ) )
138, 12imbi12d 321 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  (
x  e.  J  /\  y  e.  J )
)  ->  ( (
( ( x  i^i 
S )  =/=  (/)  /\  (
y  i^i  S )  =/=  (/)  /\  ( ( x  i^i  y )  i^i  S )  =  (/) )  ->  ( ( x  u.  y )  i^i  S )  =/= 
S )  <->  ( (
( x  i^i  S
)  =/=  (/)  /\  (
y  i^i  S )  =/=  (/)  /\  ( x  i^i  y )  C_  ( X  \  S ) )  ->  -.  S  C_  ( x  u.  y
) ) ) )
14132ralbidva 2874 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  ( A. x  e.  J  A. y  e.  J  ( ( ( x  i^i  S )  =/=  (/)  /\  ( y  i^i 
S )  =/=  (/)  /\  (
( x  i^i  y
)  i^i  S )  =  (/) )  ->  (
( x  u.  y
)  i^i  S )  =/=  S )  <->  A. x  e.  J  A. y  e.  J  ( (
( x  i^i  S
)  =/=  (/)  /\  (
y  i^i  S )  =/=  (/)  /\  ( x  i^i  y )  C_  ( X  \  S ) )  ->  -.  S  C_  ( x  u.  y
) ) ) )
151, 14bitrd 256 1  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  ->  (
( Jt  S )  e.  Con  <->  A. x  e.  J  A. y  e.  J  (
( ( x  i^i 
S )  =/=  (/)  /\  (
y  i^i  S )  =/=  (/)  /\  ( x  i^i  y )  C_  ( X  \  S ) )  ->  -.  S  C_  ( x  u.  y
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782    \ cdif 3439    u. cun 3440    i^i cin 3441    C_ wss 3442   (/)c0 3767   ` cfv 5601  (class class class)co 6305   ↾t crest 15278  TopOnctopon 19849   Conccon 20357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-oadd 7194  df-er 7371  df-en 7578  df-fin 7581  df-fi 7931  df-rest 15280  df-topgen 15301  df-top 19852  df-bases 19853  df-topon 19854  df-cld 19965  df-con 20358
This theorem is referenced by:  iuncon  20374  clscon  20376  reconn  21757  iunconlem2  36972
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