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Theorem nconsubb 19769
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 19766 . . . 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 1176 . . . 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 3698 . . . . . . . . 9  |-  ( x  =  U  ->  (
x  i^i  A )  =  ( U  i^i  A ) )
1312neeq1d 2744 . . . . . . . 8  |-  ( x  =  U  ->  (
( x  i^i  A
)  =/=  (/)  <->  ( U  i^i  A )  =/=  (/) ) )
14 ineq1 3698 . . . . . . . . . 10  |-  ( x  =  U  ->  (
x  i^i  y )  =  ( U  i^i  y ) )
1514ineq1d 3704 . . . . . . . . 9  |-  ( x  =  U  ->  (
( x  i^i  y
)  i^i  A )  =  ( ( U  i^i  y )  i^i 
A ) )
1615eqeq1d 2469 . . . . . . . 8  |-  ( x  =  U  ->  (
( ( x  i^i  y )  i^i  A
)  =  (/)  <->  ( ( U  i^i  y )  i^i 
A )  =  (/) ) )
1713, 163anbi13d 1301 . . . . . . 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 3656 . . . . . . . . 9  |-  ( x  =  U  ->  (
x  u.  y )  =  ( U  u.  y ) )
1918ineq1d 3704 . . . . . . . 8  |-  ( x  =  U  ->  (
( x  u.  y
)  i^i  A )  =  ( ( U  u.  y )  i^i 
A ) )
2019neeq1d 2744 . . . . . . 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 3698 . . . . . . . . 9  |-  ( y  =  V  ->  (
y  i^i  A )  =  ( V  i^i  A ) )
2322neeq1d 2744 . . . . . . . 8  |-  ( y  =  V  ->  (
( y  i^i  A
)  =/=  (/)  <->  ( V  i^i  A )  =/=  (/) ) )
24 ineq2 3699 . . . . . . . . . 10  |-  ( y  =  V  ->  ( U  i^i  y )  =  ( U  i^i  V
) )
2524ineq1d 3704 . . . . . . . . 9  |-  ( y  =  V  ->  (
( U  i^i  y
)  i^i  A )  =  ( ( U  i^i  V )  i^i 
A ) )
2625eqeq1d 2469 . . . . . . . 8  |-  ( y  =  V  ->  (
( ( U  i^i  y )  i^i  A
)  =  (/)  <->  ( ( U  i^i  V )  i^i 
A )  =  (/) ) )
2723, 263anbi23d 1302 . . . . . . 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 3708 . . . . . . . . 9  |-  ( A 
C_  ( U  u.  y )  <->  ( ( U  u.  y )  i^i  A )  =  A )
2928necon3bbii 2728 . . . . . . . 8  |-  ( -.  A  C_  ( U  u.  y )  <->  ( ( U  u.  y )  i^i  A )  =/=  A
)
30 uneq2 3657 . . . . . . . . . 10  |-  ( y  =  V  ->  ( U  u.  y )  =  ( U  u.  V ) )
3130sseq2d 3537 . . . . . . . . 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 3228 . . . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817    u. cun 3479    i^i cin 3480    C_ wss 3481   (/)c0 3790   ` cfv 5593  (class class class)co 6294   ↾t crest 14688  TopOnctopon 19241   Conccon 19757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-oadd 7144  df-er 7321  df-en 7527  df-fin 7530  df-fi 7881  df-rest 14690  df-topgen 14711  df-top 19245  df-bases 19247  df-topon 19248  df-cld 19365  df-con 19758
This theorem is referenced by:  iunconlem  19773  clscon  19776  reconnlem1  21176  ordtconlem1  27699
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