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Theorem nconsubb 18927
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 18924 . . . 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 656 . . 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 1163 . . . 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 3542 . . . . . . . . 9  |-  ( x  =  U  ->  (
x  i^i  A )  =  ( U  i^i  A ) )
1312neeq1d 2619 . . . . . . . 8  |-  ( x  =  U  ->  (
( x  i^i  A
)  =/=  (/)  <->  ( U  i^i  A )  =/=  (/) ) )
14 ineq1 3542 . . . . . . . . . 10  |-  ( x  =  U  ->  (
x  i^i  y )  =  ( U  i^i  y ) )
1514ineq1d 3548 . . . . . . . . 9  |-  ( x  =  U  ->  (
( x  i^i  y
)  i^i  A )  =  ( ( U  i^i  y )  i^i 
A ) )
1615eqeq1d 2449 . . . . . . . 8  |-  ( x  =  U  ->  (
( ( x  i^i  y )  i^i  A
)  =  (/)  <->  ( ( U  i^i  y )  i^i 
A )  =  (/) ) )
1713, 163anbi13d 1286 . . . . . . 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 3500 . . . . . . . . 9  |-  ( x  =  U  ->  (
x  u.  y )  =  ( U  u.  y ) )
1918ineq1d 3548 . . . . . . . 8  |-  ( x  =  U  ->  (
( x  u.  y
)  i^i  A )  =  ( ( U  u.  y )  i^i 
A ) )
2019neeq1d 2619 . . . . . . 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 3542 . . . . . . . . 9  |-  ( y  =  V  ->  (
y  i^i  A )  =  ( V  i^i  A ) )
2322neeq1d 2619 . . . . . . . 8  |-  ( y  =  V  ->  (
( y  i^i  A
)  =/=  (/)  <->  ( V  i^i  A )  =/=  (/) ) )
24 ineq2 3543 . . . . . . . . . 10  |-  ( y  =  V  ->  ( U  i^i  y )  =  ( U  i^i  V
) )
2524ineq1d 3548 . . . . . . . . 9  |-  ( y  =  V  ->  (
( U  i^i  y
)  i^i  A )  =  ( ( U  i^i  V )  i^i 
A ) )
2625eqeq1d 2449 . . . . . . . 8  |-  ( y  =  V  ->  (
( ( U  i^i  y )  i^i  A
)  =  (/)  <->  ( ( U  i^i  V )  i^i 
A )  =  (/) ) )
2723, 263anbi23d 1287 . . . . . . 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 3552 . . . . . . . . 9  |-  ( A 
C_  ( U  u.  y )  <->  ( ( U  u.  y )  i^i  A )  =  A )
2928necon3bbii 2637 . . . . . . . 8  |-  ( -.  A  C_  ( U  u.  y )  <->  ( ( U  u.  y )  i^i  A )  =/=  A
)
30 uneq2 3501 . . . . . . . . . 10  |-  ( y  =  V  ->  ( U  u.  y )  =  ( U  u.  V ) )
3130sseq2d 3381 . . . . . . . . 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 3076 . . . . 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 656 . . . 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713    u. cun 3323    i^i cin 3324    C_ wss 3325   (/)c0 3634   ` cfv 5415  (class class class)co 6090   ↾t crest 14355  TopOnctopon 18399   Conccon 18915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-oadd 6920  df-er 7097  df-en 7307  df-fin 7310  df-fi 7657  df-rest 14357  df-topgen 14378  df-top 18403  df-bases 18405  df-topon 18406  df-cld 18523  df-con 18916
This theorem is referenced by:  iunconlem  18931  clscon  18934  reconnlem1  20303  ordtconlem1  26274
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