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Theorem nconsubb 19027
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 19024 . . . 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 661 . . 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 1168 . . . 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 3545 . . . . . . . . 9  |-  ( x  =  U  ->  (
x  i^i  A )  =  ( U  i^i  A ) )
1312neeq1d 2621 . . . . . . . 8  |-  ( x  =  U  ->  (
( x  i^i  A
)  =/=  (/)  <->  ( U  i^i  A )  =/=  (/) ) )
14 ineq1 3545 . . . . . . . . . 10  |-  ( x  =  U  ->  (
x  i^i  y )  =  ( U  i^i  y ) )
1514ineq1d 3551 . . . . . . . . 9  |-  ( x  =  U  ->  (
( x  i^i  y
)  i^i  A )  =  ( ( U  i^i  y )  i^i 
A ) )
1615eqeq1d 2451 . . . . . . . 8  |-  ( x  =  U  ->  (
( ( x  i^i  y )  i^i  A
)  =  (/)  <->  ( ( U  i^i  y )  i^i 
A )  =  (/) ) )
1713, 163anbi13d 1291 . . . . . . 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 3503 . . . . . . . . 9  |-  ( x  =  U  ->  (
x  u.  y )  =  ( U  u.  y ) )
1918ineq1d 3551 . . . . . . . 8  |-  ( x  =  U  ->  (
( x  u.  y
)  i^i  A )  =  ( ( U  u.  y )  i^i 
A ) )
2019neeq1d 2621 . . . . . . 7  |-  ( x  =  U  ->  (
( ( x  u.  y )  i^i  A
)  =/=  A  <->  ( ( U  u.  y )  i^i  A )  =/=  A
) )
2117, 20imbi12d 320 . . . . . 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 3545 . . . . . . . . 9  |-  ( y  =  V  ->  (
y  i^i  A )  =  ( V  i^i  A ) )
2322neeq1d 2621 . . . . . . . 8  |-  ( y  =  V  ->  (
( y  i^i  A
)  =/=  (/)  <->  ( V  i^i  A )  =/=  (/) ) )
24 ineq2 3546 . . . . . . . . . 10  |-  ( y  =  V  ->  ( U  i^i  y )  =  ( U  i^i  V
) )
2524ineq1d 3551 . . . . . . . . 9  |-  ( y  =  V  ->  (
( U  i^i  y
)  i^i  A )  =  ( ( U  i^i  V )  i^i 
A ) )
2625eqeq1d 2451 . . . . . . . 8  |-  ( y  =  V  ->  (
( ( U  i^i  y )  i^i  A
)  =  (/)  <->  ( ( U  i^i  V )  i^i 
A )  =  (/) ) )
2723, 263anbi23d 1292 . . . . . . 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 3555 . . . . . . . . 9  |-  ( A 
C_  ( U  u.  y )  <->  ( ( U  u.  y )  i^i  A )  =  A )
2928necon3bbii 2639 . . . . . . . 8  |-  ( -.  A  C_  ( U  u.  y )  <->  ( ( U  u.  y )  i^i  A )  =/=  A
)
30 uneq2 3504 . . . . . . . . . 10  |-  ( y  =  V  ->  ( U  u.  y )  =  ( U  u.  V ) )
3130sseq2d 3384 . . . . . . . . 9  |-  ( y  =  V  ->  ( A  C_  ( U  u.  y )  <->  A  C_  ( U  u.  V )
) )
3231notbid 294 . . . . . . . 8  |-  ( y  =  V  ->  ( -.  A  C_  ( U  u.  y )  <->  -.  A  C_  ( U  u.  V
) ) )
3329, 32syl5bbr 259 . . . . . . 7  |-  ( y  =  V  ->  (
( ( U  u.  y )  i^i  A
)  =/=  A  <->  -.  A  C_  ( U  u.  V
) ) )
3427, 33imbi12d 320 . . . . . 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 3079 . . . . 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 661 . . . 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 41 . . 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 215 . 2  |-  ( ph  ->  ( ( Jt  A )  e.  Con  ->  -.  A  C_  ( U  u.  V ) ) )
391, 38mt2d 117 1  |-  ( ph  ->  -.  ( Jt  A )  e.  Con )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715    u. cun 3326    i^i cin 3327    C_ wss 3328   (/)c0 3637   ` cfv 5418  (class class class)co 6091   ↾t crest 14359  TopOnctopon 18499   Conccon 19015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-oadd 6924  df-er 7101  df-en 7311  df-fin 7314  df-fi 7661  df-rest 14361  df-topgen 14382  df-top 18503  df-bases 18505  df-topon 18506  df-cld 18623  df-con 19016
This theorem is referenced by:  iunconlem  19031  clscon  19034  reconnlem1  20403  ordtconlem1  26354
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