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Theorem uncon 17155
Description: The union of two connected overlapping subspaces is connected. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 11-Jun-2014.)
Assertion
Ref Expression
uncon  |-  ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con )  ->  ( Jt  ( A  u.  B ) )  e.  Con ) )

Proof of Theorem uncon
Dummy variables  x  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3464 . . 3  |-  ( ( A  i^i  B )  =/=  (/)  <->  E. x  x  e.  ( A  i^i  B
) )
2 uniiun 3955 . . . . . . . . 9  |-  U. { A ,  B }  =  U_ k  e.  { A ,  B }
k
3 simpl2l 1008 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  A  C_  X )
4 simpl1 958 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  J  e.  (TopOn `  X )
)
5 toponmax 16666 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
64, 5syl 15 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  X  e.  J )
7 ssexg 4160 . . . . . . . . . . 11  |-  ( ( A  C_  X  /\  X  e.  J )  ->  A  e.  _V )
83, 6, 7syl2anc 642 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  A  e.  _V )
9 simpl2r 1009 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  B  C_  X )
10 ssexg 4160 . . . . . . . . . . 11  |-  ( ( B  C_  X  /\  X  e.  J )  ->  B  e.  _V )
119, 6, 10syl2anc 642 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  B  e.  _V )
12 uniprg 3842 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  U. { A ,  B }  =  ( A  u.  B )
)
138, 11, 12syl2anc 642 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  U. { A ,  B }  =  ( A  u.  B ) )
142, 13syl5eqr 2329 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  U_ k  e.  { A ,  B } k  =  ( A  u.  B ) )
1514oveq2d 5874 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  ( Jt  U_ k  e.  { A ,  B } k )  =  ( Jt  ( A  u.  B ) ) )
16 vex 2791 . . . . . . . . . 10  |-  k  e. 
_V
1716elpr 3658 . . . . . . . . 9  |-  ( k  e.  { A ,  B }  <->  ( k  =  A  \/  k  =  B ) )
18 simpl2 959 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  ( A  C_  X  /\  B  C_  X ) )
19 sseq1 3199 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
k  C_  X  <->  A  C_  X
) )
2019biimprd 214 . . . . . . . . . . 11  |-  ( k  =  A  ->  ( A  C_  X  ->  k  C_  X ) )
21 sseq1 3199 . . . . . . . . . . . 12  |-  ( k  =  B  ->  (
k  C_  X  <->  B  C_  X
) )
2221biimprd 214 . . . . . . . . . . 11  |-  ( k  =  B  ->  ( B  C_  X  ->  k  C_  X ) )
2320, 22jaoa 496 . . . . . . . . . 10  |-  ( ( k  =  A  \/  k  =  B )  ->  ( ( A  C_  X  /\  B  C_  X
)  ->  k  C_  X ) )
2418, 23mpan9 455 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B ) )  /\  ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  /\  (
k  =  A  \/  k  =  B )
)  ->  k  C_  X )
2517, 24sylan2b 461 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B ) )  /\  ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  /\  k  e.  { A ,  B } )  ->  k  C_  X )
26 simpl3 960 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  x  e.  ( A  i^i  B
) )
27 elin 3358 . . . . . . . . . . 11  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
2826, 27sylib 188 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  (
x  e.  A  /\  x  e.  B )
)
29 eleq2 2344 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
x  e.  k  <->  x  e.  A ) )
3029biimprd 214 . . . . . . . . . . 11  |-  ( k  =  A  ->  (
x  e.  A  ->  x  e.  k )
)
31 eleq2 2344 . . . . . . . . . . . 12  |-  ( k  =  B  ->  (
x  e.  k  <->  x  e.  B ) )
3231biimprd 214 . . . . . . . . . . 11  |-  ( k  =  B  ->  (
x  e.  B  ->  x  e.  k )
)
3330, 32jaoa 496 . . . . . . . . . 10  |-  ( ( k  =  A  \/  k  =  B )  ->  ( ( x  e.  A  /\  x  e.  B )  ->  x  e.  k ) )
3428, 33mpan9 455 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B ) )  /\  ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  /\  (
k  =  A  \/  k  =  B )
)  ->  x  e.  k )
3517, 34sylan2b 461 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B ) )  /\  ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  /\  k  e.  { A ,  B } )  ->  x  e.  k )
36 simpr 447 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )
37 oveq2 5866 . . . . . . . . . . . . 13  |-  ( k  =  A  ->  ( Jt  k )  =  ( Jt  A ) )
3837eleq1d 2349 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
( Jt  k )  e. 
Con 
<->  ( Jt  A )  e.  Con ) )
3938biimprd 214 . . . . . . . . . . 11  |-  ( k  =  A  ->  (
( Jt  A )  e.  Con  ->  ( Jt  k )  e. 
Con ) )
40 oveq2 5866 . . . . . . . . . . . . 13  |-  ( k  =  B  ->  ( Jt  k )  =  ( Jt  B ) )
4140eleq1d 2349 . . . . . . . . . . . 12  |-  ( k  =  B  ->  (
( Jt  k )  e. 
Con 
<->  ( Jt  B )  e.  Con ) )
4241biimprd 214 . . . . . . . . . . 11  |-  ( k  =  B  ->  (
( Jt  B )  e.  Con  ->  ( Jt  k )  e. 
Con ) )
4339, 42jaoa 496 . . . . . . . . . 10  |-  ( ( k  =  A  \/  k  =  B )  ->  ( ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con )  ->  ( Jt  k )  e.  Con ) )
4436, 43mpan9 455 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B ) )  /\  ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  /\  (
k  =  A  \/  k  =  B )
)  ->  ( Jt  k
)  e.  Con )
4517, 44sylan2b 461 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B ) )  /\  ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  /\  k  e.  { A ,  B } )  ->  ( Jt  k )  e.  Con )
464, 25, 35, 45iuncon 17154 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  ( Jt  U_ k  e.  { A ,  B } k )  e.  Con )
4715, 46eqeltrrd 2358 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  /\  (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con ) )  ->  ( Jt  ( A  u.  B
) )  e.  Con )
4847ex 423 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  x  e.  ( A  i^i  B
) )  ->  (
( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con )  ->  ( Jt  ( A  u.  B ) )  e.  Con ) )
49483expia 1153 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X ) )  -> 
( x  e.  ( A  i^i  B )  ->  ( ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con )  ->  ( Jt  ( A  u.  B ) )  e.  Con ) ) )
5049exlimdv 1664 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X ) )  -> 
( E. x  x  e.  ( A  i^i  B )  ->  ( (
( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con )  ->  ( Jt  ( A  u.  B ) )  e.  Con ) ) )
511, 50syl5bi 208 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X ) )  -> 
( ( A  i^i  B )  =/=  (/)  ->  (
( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con )  ->  ( Jt  ( A  u.  B ) )  e.  Con ) ) )
52513impia 1148 1  |-  ( ( J  e.  (TopOn `  X )  /\  ( A  C_  X  /\  B  C_  X )  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con )  ->  ( Jt  ( A  u.  B ) )  e.  Con ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {cpr 3641   U.cuni 3827   U_ciun 3905   ` cfv 5255  (class class class)co 5858   ↾t crest 13325  TopOnctopon 16632   Conccon 17137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-con 17138
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