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Theorem uncon 17484
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 3629 . . 3  |-  ( ( A  i^i  B )  =/=  (/)  <->  E. x  x  e.  ( A  i^i  B
) )
2 uniiun 4136 . . . . . . . . 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 16985 . . . . . . . . . . . 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 4342 . . . . . . . . . 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 4342 . . . . . . . . . 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 4022 . . . . . . . . . 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 2481 . . . . . . . 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 6089 . . . . . . 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 2951 . . . . . . . . . 10  |-  k  e. 
_V
1514elpr 3824 . . . . . . . . 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 3361 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
k  C_  X  <->  A  C_  X
) )
1817biimprd 215 . . . . . . . . . . 11  |-  ( k  =  A  ->  ( A  C_  X  ->  k  C_  X ) )
19 sseq1 3361 . . . . . . . . . . . 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 3522 . . . . . . . . . . 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 2496 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
x  e.  k  <->  x  e.  A ) )
2827biimprd 215 . . . . . . . . . . 11  |-  ( k  =  A  ->  (
x  e.  A  ->  x  e.  k )
)
29 eleq2 2496 . . . . . . . . . . . 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 6081 . . . . . . . . . . . . 13  |-  ( k  =  A  ->  ( Jt  k )  =  ( Jt  A ) )
3635eleq1d 2501 . . . . . . . . . . . 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 6081 . . . . . . . . . . . . 13  |-  ( k  =  B  ->  ( Jt  k )  =  ( Jt  B ) )
3938eleq1d 2501 . . . . . . . . . . . 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 17483 . . . . . . 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 2510 . . . . . 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 1646 . . 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 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   {cpr 3807   U.cuni 4007   U_ciun 4085   ` cfv 5446  (class class class)co 6073   ↾t crest 13640  TopOnctopon 16951   Conccon 17466
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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