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Theorem uncon 20381
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 3768 . . 3  |-  ( ( A  i^i  B )  =/=  (/)  <->  E. x  x  e.  ( A  i^i  B
) )
2 uniiun 4346 . . . . . . . . 9  |-  U. { A ,  B }  =  U_ k  e.  { A ,  B }
k
3 simpl1 1008 . . . . . . . . . . . 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 19880 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
53, 4syl 17 . . . . . . . . . . 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 1058 . . . . . . . . . . 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 4563 . . . . . . . . . 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 1059 . . . . . . . . . . 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 4563 . . . . . . . . . 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 4227 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  U. { A ,  B }  =  ( A  u.  B )
)
117, 9, 10syl2anc 665 . . . . . . . . 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 2475 . . . . . . . 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 6312 . . . . . . 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 3081 . . . . . . . . . 10  |-  k  e. 
_V
1514elpr 4011 . . . . . . . . 9  |-  ( k  e.  { A ,  B }  <->  ( k  =  A  \/  k  =  B ) )
16 simpl2 1009 . . . . . . . . . 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 3482 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
k  C_  X  <->  A  C_  X
) )
1817biimprd 226 . . . . . . . . . . 11  |-  ( k  =  A  ->  ( A  C_  X  ->  k  C_  X ) )
19 sseq1 3482 . . . . . . . . . . . 12  |-  ( k  =  B  ->  (
k  C_  X  <->  B  C_  X
) )
2019biimprd 226 . . . . . . . . . . 11  |-  ( k  =  B  ->  ( B  C_  X  ->  k  C_  X ) )
2118, 20jaoa 512 . . . . . . . . . 10  |-  ( ( k  =  A  \/  k  =  B )  ->  ( ( A  C_  X  /\  B  C_  X
)  ->  k  C_  X ) )
2216, 21mpan9 471 . . . . . . . . 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 477 . . . . . . . 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 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 ) )  ->  x  e.  ( A  i^i  B
) )
25 elin 3646 . . . . . . . . . . 11  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
2624, 25sylib 199 . . . . . . . . . 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 2493 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
x  e.  k  <->  x  e.  A ) )
2827biimprd 226 . . . . . . . . . . 11  |-  ( k  =  A  ->  (
x  e.  A  ->  x  e.  k )
)
29 eleq2 2493 . . . . . . . . . . . 12  |-  ( k  =  B  ->  (
x  e.  k  <->  x  e.  B ) )
3029biimprd 226 . . . . . . . . . . 11  |-  ( k  =  B  ->  (
x  e.  B  ->  x  e.  k )
)
3128, 30jaoa 512 . . . . . . . . . 10  |-  ( ( k  =  A  \/  k  =  B )  ->  ( ( x  e.  A  /\  x  e.  B )  ->  x  e.  k ) )
3226, 31mpan9 471 . . . . . . . . 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 477 . . . . . . . 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 462 . . . . . . . . . 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 6304 . . . . . . . . . . . . 13  |-  ( k  =  A  ->  ( Jt  k )  =  ( Jt  A ) )
3635eleq1d 2489 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
( Jt  k )  e. 
Con 
<->  ( Jt  A )  e.  Con ) )
3736biimprd 226 . . . . . . . . . . 11  |-  ( k  =  A  ->  (
( Jt  A )  e.  Con  ->  ( Jt  k )  e. 
Con ) )
38 oveq2 6304 . . . . . . . . . . . . 13  |-  ( k  =  B  ->  ( Jt  k )  =  ( Jt  B ) )
3938eleq1d 2489 . . . . . . . . . . . 12  |-  ( k  =  B  ->  (
( Jt  k )  e. 
Con 
<->  ( Jt  B )  e.  Con ) )
4039biimprd 226 . . . . . . . . . . 11  |-  ( k  =  B  ->  (
( Jt  B )  e.  Con  ->  ( Jt  k )  e. 
Con ) )
4137, 40jaoa 512 . . . . . . . . . 10  |-  ( ( k  =  A  \/  k  =  B )  ->  ( ( ( Jt  A )  e.  Con  /\  ( Jt  B )  e.  Con )  ->  ( Jt  k )  e.  Con ) )
4234, 41mpan9 471 . . . . . . . . 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 477 . . . . . . . 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 20380 . . . . . . 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 2509 . . . . . 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 435 . . . . 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 1207 . . . 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 1768 . . 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 220 . 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 1202 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 setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1867    =/= wne 2616   _Vcvv 3078    u. cun 3431    i^i cin 3432    C_ wss 3433   (/)c0 3758   {cpr 3995   U.cuni 4213   U_ciun 4293   ` cfv 5592  (class class class)co 6296   ↾t crest 15279  TopOnctopon 19855   Conccon 20363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-oadd 7185  df-er 7362  df-en 7569  df-fin 7572  df-fi 7922  df-rest 15281  df-topgen 15302  df-top 19858  df-bases 19859  df-topon 19860  df-cld 19971  df-con 20364
This theorem is referenced by: (None)
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