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Theorem uncon 17414
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 3581 . . 3  |-  ( ( A  i^i  B )  =/=  (/)  <->  E. x  x  e.  ( A  i^i  B
) )
2 uniiun 4086 . . . . . . . . 9  |-  U. { A ,  B }  =  U_ k  e.  { A ,  B }
k
3 simpl1 960 . . . . . . . . . . . 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 )
)
4 toponmax 16917 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
53, 4syl 16 . . . . . . . . . . 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 )
6 simpl2l 1010 . . . . . . . . . . 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 )
75, 6ssexd 4292 . . . . . . . . . 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 )
8 simpl2r 1011 . . . . . . . . . . 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 )
95, 8ssexd 4292 . . . . . . . . . 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 )
10 uniprg 3973 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  U. { A ,  B }  =  ( A  u.  B )
)
117, 9, 10syl2anc 643 . . . . . . . . 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 ) )
122, 11syl5eqr 2434 . . . . . . . 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 ) )
1312oveq2d 6037 . . . . . . 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 ) ) )
14 vex 2903 . . . . . . . . . 10  |-  k  e. 
_V
1514elpr 3776 . . . . . . . . 9  |-  ( k  e.  { A ,  B }  <->  ( k  =  A  \/  k  =  B ) )
16 simpl2 961 . . . . . . . . . 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 ) )
17 sseq1 3313 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
k  C_  X  <->  A  C_  X
) )
1817biimprd 215 . . . . . . . . . . 11  |-  ( k  =  A  ->  ( A  C_  X  ->  k  C_  X ) )
19 sseq1 3313 . . . . . . . . . . . 12  |-  ( k  =  B  ->  (
k  C_  X  <->  B  C_  X
) )
2019biimprd 215 . . . . . . . . . . 11  |-  ( k  =  B  ->  ( B  C_  X  ->  k  C_  X ) )
2118, 20jaoa 497 . . . . . . . . . 10  |-  ( ( k  =  A  \/  k  =  B )  ->  ( ( A  C_  X  /\  B  C_  X
)  ->  k  C_  X ) )
2216, 21mpan9 456 . . . . . . . . 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 )
2315, 22sylan2b 462 . . . . . . . 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 )
24 simpl3 962 . . . . . . . . . . 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
) )
25 elin 3474 . . . . . . . . . . 11  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
2624, 25sylib 189 . . . . . . . . . 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 )
)
27 eleq2 2449 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
x  e.  k  <->  x  e.  A ) )
2827biimprd 215 . . . . . . . . . . 11  |-  ( k  =  A  ->  (
x  e.  A  ->  x  e.  k )
)
29 eleq2 2449 . . . . . . . . . . . 12  |-  ( k  =  B  ->  (
x  e.  k  <->  x  e.  B ) )
3029biimprd 215 . . . . . . . . . . 11  |-  ( k  =  B  ->  (
x  e.  B  ->  x  e.  k )
)
3128, 30jaoa 497 . . . . . . . . . 10  |-  ( ( k  =  A  \/  k  =  B )  ->  ( ( x  e.  A  /\  x  e.  B )  ->  x  e.  k ) )
3226, 31mpan9 456 . . . . . . . . 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 )
3315, 32sylan2b 462 . . . . . . . 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 )
34 simpr 448 . . . . . . . . . 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 ) )
35 oveq2 6029 . . . . . . . . . . . . 13  |-  ( k  =  A  ->  ( Jt  k )  =  ( Jt  A ) )
3635eleq1d 2454 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
( Jt  k )  e. 
Con 
<->  ( Jt  A )  e.  Con ) )
3736biimprd 215 . . . . . . . . . . 11  |-  ( k  =  A  ->  (
( Jt  A )  e.  Con  ->  ( Jt  k )  e. 
Con ) )
38 oveq2 6029 . . . . . . . . . . . . 13  |-  ( k  =  B  ->  ( Jt  k )  =  ( Jt  B ) )
3938eleq1d 2454 . . . . . . . . . . . 12  |-  ( k  =  B  ->  (
( Jt  k )  e. 
Con 
<->  ( Jt  B )  e.  Con ) )
4039biimprd 215 . . . . . . . . . . 11  |-  ( k  =  B  ->  (
( Jt  B )  e.  Con  ->  ( Jt  k )  e. 
Con ) )
4137, 40jaoa 497 . . . . . . . . . 10  |-  ( ( k  =  A  \/  k  =  B )  ->  ( ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con )  ->  ( Jt  k )  e.  Con ) )
4234, 41mpan9 456 . . . . . . . . 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 )
4315, 42sylan2b 462 . . . . . . . 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 )
443, 23, 33, 43iuncon 17413 . . . . . . 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 )
4513, 44eqeltrrd 2463 . . . . . 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 )
4645ex 424 . . . . 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 ) )
47463expia 1155 . . . 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 ) ) )
4847exlimdv 1643 . . 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 ) ) )
491, 48syl5bi 209 . 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 ) ) )
50493impia 1150 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 358    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900    u. cun 3262    i^i cin 3263    C_ wss 3264   (/)c0 3572   {cpr 3759   U.cuni 3958   U_ciun 4036   ` cfv 5395  (class class class)co 6021   ↾t crest 13576  TopOnctopon 16883   Conccon 17396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-recs 6570  df-rdg 6605  df-oadd 6665  df-er 6842  df-en 7047  df-fin 7050  df-fi 7352  df-rest 13578  df-topgen 13595  df-top 16887  df-bases 16889  df-topon 16890  df-cld 17007  df-con 17397
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