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Theorem nconsubb 17478
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 17475 . . . 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 643 . . 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 1134 . . . 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 3527 . . . . . . . . 9  |-  ( x  =  U  ->  (
x  i^i  A )  =  ( U  i^i  A ) )
1312neeq1d 2611 . . . . . . . 8  |-  ( x  =  U  ->  (
( x  i^i  A
)  =/=  (/)  <->  ( U  i^i  A )  =/=  (/) ) )
14 ineq1 3527 . . . . . . . . . 10  |-  ( x  =  U  ->  (
x  i^i  y )  =  ( U  i^i  y ) )
1514ineq1d 3533 . . . . . . . . 9  |-  ( x  =  U  ->  (
( x  i^i  y
)  i^i  A )  =  ( ( U  i^i  y )  i^i 
A ) )
1615eqeq1d 2443 . . . . . . . 8  |-  ( x  =  U  ->  (
( ( x  i^i  y )  i^i  A
)  =  (/)  <->  ( ( U  i^i  y )  i^i 
A )  =  (/) ) )
1713, 163anbi13d 1256 . . . . . . 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 3486 . . . . . . . . 9  |-  ( x  =  U  ->  (
x  u.  y )  =  ( U  u.  y ) )
1918ineq1d 3533 . . . . . . . 8  |-  ( x  =  U  ->  (
( x  u.  y
)  i^i  A )  =  ( ( U  u.  y )  i^i 
A ) )
2019neeq1d 2611 . . . . . . 7  |-  ( x  =  U  ->  (
( ( x  u.  y )  i^i  A
)  =/=  A  <->  ( ( U  u.  y )  i^i  A )  =/=  A
) )
2117, 20imbi12d 312 . . . . . 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 3527 . . . . . . . . 9  |-  ( y  =  V  ->  (
y  i^i  A )  =  ( V  i^i  A ) )
2322neeq1d 2611 . . . . . . . 8  |-  ( y  =  V  ->  (
( y  i^i  A
)  =/=  (/)  <->  ( V  i^i  A )  =/=  (/) ) )
24 ineq2 3528 . . . . . . . . . 10  |-  ( y  =  V  ->  ( U  i^i  y )  =  ( U  i^i  V
) )
2524ineq1d 3533 . . . . . . . . 9  |-  ( y  =  V  ->  (
( U  i^i  y
)  i^i  A )  =  ( ( U  i^i  V )  i^i 
A ) )
2625eqeq1d 2443 . . . . . . . 8  |-  ( y  =  V  ->  (
( ( U  i^i  y )  i^i  A
)  =  (/)  <->  ( ( U  i^i  V )  i^i 
A )  =  (/) ) )
2723, 263anbi23d 1257 . . . . . . 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 3537 . . . . . . . . 9  |-  ( A 
C_  ( U  u.  y )  <->  ( ( U  u.  y )  i^i  A )  =  A )
2928necon3bbii 2629 . . . . . . . 8  |-  ( -.  A  C_  ( U  u.  y )  <->  ( ( U  u.  y )  i^i  A )  =/=  A
)
30 uneq2 3487 . . . . . . . . . 10  |-  ( y  =  V  ->  ( U  u.  y )  =  ( U  u.  V ) )
3130sseq2d 3368 . . . . . . . . 9  |-  ( y  =  V  ->  ( A  C_  ( U  u.  y )  <->  A  C_  ( U  u.  V )
) )
3231notbid 286 . . . . . . . 8  |-  ( y  =  V  ->  ( -.  A  C_  ( U  u.  y )  <->  -.  A  C_  ( U  u.  V
) ) )
3329, 32syl5bbr 251 . . . . . . 7  |-  ( y  =  V  ->  (
( ( U  u.  y )  i^i  A
)  =/=  A  <->  -.  A  C_  ( U  u.  V
) ) )
3427, 33imbi12d 312 . . . . . 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 3050 . . . . 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 643 . . . 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 39 . . 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 207 . 2  |-  ( ph  ->  ( ( Jt  A )  e.  Con  ->  -.  A  C_  ( U  u.  V ) ) )
391, 38mt2d 111 1  |-  ( ph  ->  -.  ( Jt  A )  e.  Con )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   ` cfv 5446  (class class class)co 6073   ↾t crest 13640  TopOnctopon 16951   Conccon 17466
This theorem is referenced by:  iunconlem  17482  clscon  17485  reconnlem1  18849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-oadd 6720  df-er 6897  df-en 7102  df-fin 7105  df-fi 7408  df-rest 13642  df-topgen 13659  df-top 16955  df-bases 16957  df-topon 16958  df-cld 17075  df-con 17467
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