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Theorem nconsubb 17165
Description: Disconnectedness for a subspace. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
Hypotheses
Ref Expression
nconsubb.2  |-  ( ph  ->  J  e.  (TopOn `  X ) )
nconsubb.3  |-  ( ph  ->  A  C_  X )
nconsubb.4  |-  ( ph  ->  U  e.  J )
nconsubb.5  |-  ( ph  ->  V  e.  J )
nconsubb.6  |-  ( ph  ->  ( U  i^i  A
)  =/=  (/) )
nconsubb.7  |-  ( ph  ->  ( V  i^i  A
)  =/=  (/) )
nconsubb.8  |-  ( ph  ->  ( ( U  i^i  V )  i^i  A )  =  (/) )
nconsubb.9  |-  ( ph  ->  A  C_  ( U  u.  V ) )
Assertion
Ref Expression
nconsubb  |-  ( ph  ->  -.  ( Jt  A )  e.  Con )

Proof of Theorem nconsubb
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nconsubb.9 . 2  |-  ( ph  ->  A  C_  ( U  u.  V ) )
2 nconsubb.2 . . . 4  |-  ( ph  ->  J  e.  (TopOn `  X ) )
3 nconsubb.3 . . . 4  |-  ( ph  ->  A  C_  X )
4 consuba 17162 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
( Jt  A )  e.  Con  <->  A. x  e.  J  A. y  e.  J  (
( ( x  i^i 
A )  =/=  (/)  /\  (
y  i^i  A )  =/=  (/)  /\  ( ( x  i^i  y )  i^i  A )  =  (/) )  ->  ( ( x  u.  y )  i^i  A )  =/= 
A ) ) )
52, 3, 4syl2anc 642 . . 3  |-  ( ph  ->  ( ( Jt  A )  e.  Con  <->  A. x  e.  J  A. y  e.  J  ( (
( x  i^i  A
)  =/=  (/)  /\  (
y  i^i  A )  =/=  (/)  /\  ( ( x  i^i  y )  i^i  A )  =  (/) )  ->  ( ( x  u.  y )  i^i  A )  =/= 
A ) ) )
6 nconsubb.6 . . . . 5  |-  ( ph  ->  ( U  i^i  A
)  =/=  (/) )
7 nconsubb.7 . . . . 5  |-  ( ph  ->  ( V  i^i  A
)  =/=  (/) )
8 nconsubb.8 . . . . 5  |-  ( ph  ->  ( ( U  i^i  V )  i^i  A )  =  (/) )
96, 7, 83jca 1132 . . . 4  |-  ( ph  ->  ( ( U  i^i  A )  =/=  (/)  /\  ( V  i^i  A )  =/=  (/)  /\  ( ( U  i^i  V )  i^i 
A )  =  (/) ) )
10 nconsubb.4 . . . . 5  |-  ( ph  ->  U  e.  J )
11 nconsubb.5 . . . . 5  |-  ( ph  ->  V  e.  J )
12 ineq1 3376 . . . . . . . . 9  |-  ( x  =  U  ->  (
x  i^i  A )  =  ( U  i^i  A ) )
1312neeq1d 2472 . . . . . . . 8  |-  ( x  =  U  ->  (
( x  i^i  A
)  =/=  (/)  <->  ( U  i^i  A )  =/=  (/) ) )
14 ineq1 3376 . . . . . . . . . 10  |-  ( x  =  U  ->  (
x  i^i  y )  =  ( U  i^i  y ) )
1514ineq1d 3382 . . . . . . . . 9  |-  ( x  =  U  ->  (
( x  i^i  y
)  i^i  A )  =  ( ( U  i^i  y )  i^i 
A ) )
1615eqeq1d 2304 . . . . . . . 8  |-  ( x  =  U  ->  (
( ( x  i^i  y )  i^i  A
)  =  (/)  <->  ( ( U  i^i  y )  i^i 
A )  =  (/) ) )
1713, 163anbi13d 1254 . . . . . . 7  |-  ( x  =  U  ->  (
( ( x  i^i 
A )  =/=  (/)  /\  (
y  i^i  A )  =/=  (/)  /\  ( ( x  i^i  y )  i^i  A )  =  (/) )  <->  ( ( U  i^i  A )  =/=  (/)  /\  ( y  i^i 
A )  =/=  (/)  /\  (
( U  i^i  y
)  i^i  A )  =  (/) ) ) )
18 uneq1 3335 . . . . . . . . 9  |-  ( x  =  U  ->  (
x  u.  y )  =  ( U  u.  y ) )
1918ineq1d 3382 . . . . . . . 8  |-  ( x  =  U  ->  (
( x  u.  y
)  i^i  A )  =  ( ( U  u.  y )  i^i 
A ) )
2019neeq1d 2472 . . . . . . 7  |-  ( x  =  U  ->  (
( ( x  u.  y )  i^i  A
)  =/=  A  <->  ( ( U  u.  y )  i^i  A )  =/=  A
) )
2117, 20imbi12d 311 . . . . . 6  |-  ( x  =  U  ->  (
( ( ( x  i^i  A )  =/=  (/)  /\  ( y  i^i 
A )  =/=  (/)  /\  (
( x  i^i  y
)  i^i  A )  =  (/) )  ->  (
( x  u.  y
)  i^i  A )  =/=  A )  <->  ( (
( U  i^i  A
)  =/=  (/)  /\  (
y  i^i  A )  =/=  (/)  /\  ( ( U  i^i  y )  i^i  A )  =  (/) )  ->  ( ( U  u.  y )  i^i  A )  =/= 
A ) ) )
22 ineq1 3376 . . . . . . . . 9  |-  ( y  =  V  ->  (
y  i^i  A )  =  ( V  i^i  A ) )
2322neeq1d 2472 . . . . . . . 8  |-  ( y  =  V  ->  (
( y  i^i  A
)  =/=  (/)  <->  ( V  i^i  A )  =/=  (/) ) )
24 ineq2 3377 . . . . . . . . . 10  |-  ( y  =  V  ->  ( U  i^i  y )  =  ( U  i^i  V
) )
2524ineq1d 3382 . . . . . . . . 9  |-  ( y  =  V  ->  (
( U  i^i  y
)  i^i  A )  =  ( ( U  i^i  V )  i^i 
A ) )
2625eqeq1d 2304 . . . . . . . 8  |-  ( y  =  V  ->  (
( ( U  i^i  y )  i^i  A
)  =  (/)  <->  ( ( U  i^i  V )  i^i 
A )  =  (/) ) )
2723, 263anbi23d 1255 . . . . . . 7  |-  ( y  =  V  ->  (
( ( U  i^i  A )  =/=  (/)  /\  (
y  i^i  A )  =/=  (/)  /\  ( ( U  i^i  y )  i^i  A )  =  (/) )  <->  ( ( U  i^i  A )  =/=  (/)  /\  ( V  i^i  A )  =/=  (/)  /\  (
( U  i^i  V
)  i^i  A )  =  (/) ) ) )
28 dfss1 3386 . . . . . . . . 9  |-  ( A 
C_  ( U  u.  y )  <->  ( ( U  u.  y )  i^i  A )  =  A )
2928necon3bbii 2490 . . . . . . . 8  |-  ( -.  A  C_  ( U  u.  y )  <->  ( ( U  u.  y )  i^i  A )  =/=  A
)
30 uneq2 3336 . . . . . . . . . 10  |-  ( y  =  V  ->  ( U  u.  y )  =  ( U  u.  V ) )
3130sseq2d 3219 . . . . . . . . 9  |-  ( y  =  V  ->  ( A  C_  ( U  u.  y )  <->  A  C_  ( U  u.  V )
) )
3231notbid 285 . . . . . . . 8  |-  ( y  =  V  ->  ( -.  A  C_  ( U  u.  y )  <->  -.  A  C_  ( U  u.  V
) ) )
3329, 32syl5bbr 250 . . . . . . 7  |-  ( y  =  V  ->  (
( ( U  u.  y )  i^i  A
)  =/=  A  <->  -.  A  C_  ( U  u.  V
) ) )
3427, 33imbi12d 311 . . . . . 6  |-  ( y  =  V  ->  (
( ( ( U  i^i  A )  =/=  (/)  /\  ( y  i^i 
A )  =/=  (/)  /\  (
( U  i^i  y
)  i^i  A )  =  (/) )  ->  (
( U  u.  y
)  i^i  A )  =/=  A )  <->  ( (
( U  i^i  A
)  =/=  (/)  /\  ( V  i^i  A )  =/=  (/)  /\  ( ( U  i^i  V )  i^i 
A )  =  (/) )  ->  -.  A  C_  ( U  u.  V )
) ) )
3521, 34rspc2v 2903 . . . . 5  |-  ( ( U  e.  J  /\  V  e.  J )  ->  ( A. x  e.  J  A. y  e.  J  ( ( ( x  i^i  A )  =/=  (/)  /\  ( y  i^i  A )  =/=  (/)  /\  ( ( x  i^i  y )  i^i 
A )  =  (/) )  ->  ( ( x  u.  y )  i^i 
A )  =/=  A
)  ->  ( (
( U  i^i  A
)  =/=  (/)  /\  ( V  i^i  A )  =/=  (/)  /\  ( ( U  i^i  V )  i^i 
A )  =  (/) )  ->  -.  A  C_  ( U  u.  V )
) ) )
3610, 11, 35syl2anc 642 . . . 4  |-  ( ph  ->  ( A. x  e.  J  A. y  e.  J  ( ( ( x  i^i  A )  =/=  (/)  /\  ( y  i^i  A )  =/=  (/)  /\  ( ( x  i^i  y )  i^i 
A )  =  (/) )  ->  ( ( x  u.  y )  i^i 
A )  =/=  A
)  ->  ( (
( U  i^i  A
)  =/=  (/)  /\  ( V  i^i  A )  =/=  (/)  /\  ( ( U  i^i  V )  i^i 
A )  =  (/) )  ->  -.  A  C_  ( U  u.  V )
) ) )
379, 36mpid 37 . . 3  |-  ( ph  ->  ( A. x  e.  J  A. y  e.  J  ( ( ( x  i^i  A )  =/=  (/)  /\  ( y  i^i  A )  =/=  (/)  /\  ( ( x  i^i  y )  i^i 
A )  =  (/) )  ->  ( ( x  u.  y )  i^i 
A )  =/=  A
)  ->  -.  A  C_  ( U  u.  V
) ) )
385, 37sylbid 206 . 2  |-  ( ph  ->  ( ( Jt  A )  e.  Con  ->  -.  A  C_  ( U  u.  V ) ) )
391, 38mt2d 109 1  |-  ( ph  ->  -.  ( Jt  A )  e.  Con )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   ` cfv 5271  (class class class)co 5874   ↾t crest 13341  TopOnctopon 16648   Conccon 17153
This theorem is referenced by:  iunconlem  17169  clscon  17172  reconnlem1  18347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-en 6880  df-fin 6883  df-fi 7181  df-rest 13343  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-con 17154
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