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Theorem connsub 17486
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 17485 . 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 3563 . . . . . . 7  |-  ( x  i^i  y )  C_  x
3 toponss 16996 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
43ad2ant2r 729 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  (
x  e.  J  /\  y  e.  J )
)  ->  x  C_  X
)
52, 4syl5ss 3361 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  S  C_  X )  /\  (
x  e.  J  /\  y  e.  J )
)  ->  ( x  i^i  y )  C_  X
)
6 reldisj 3673 . . . . . 6  |-  ( ( x  i^i  y ) 
C_  X  ->  (
( ( x  i^i  y )  i^i  S
)  =  (/)  <->  ( x  i^i  y )  C_  ( X  \  S ) ) )
75, 6syl 16 . . . . 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 1261 . . . 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 3547 . . . . . . 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 194 . . . . 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 2636 . . . 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 313 . . 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 2747 . 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 246 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   ` cfv 5456  (class class class)co 6083   ↾t crest 13650  TopOnctopon 16961   Conccon 17476
This theorem is referenced by:  iuncon  17493  clscon  17495  reconn  18861  iunconlem2  29109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-recs 6635  df-rdg 6670  df-oadd 6730  df-er 6907  df-en 7112  df-fin 7115  df-fi 7418  df-rest 13652  df-topgen 13669  df-top 16965  df-bases 16967  df-topon 16968  df-cld 17085  df-con 17477
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