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Theorem uncon 17171
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 3477 . . 3  |-  ( ( A  i^i  B )  =/=  (/)  <->  E. x  x  e.  ( A  i^i  B
) )
2 uniiun 3971 . . . . . . . . 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 16682 . . . . . . . . . . . 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 4176 . . . . . . . . . . 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 4176 . . . . . . . . . . 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 3858 . . . . . . . . . 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 2342 . . . . . . . 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 5890 . . . . . . 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 2804 . . . . . . . . . 10  |-  k  e. 
_V
1716elpr 3671 . . . . . . . . 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 3212 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
k  C_  X  <->  A  C_  X
) )
2019biimprd 214 . . . . . . . . . . 11  |-  ( k  =  A  ->  ( A  C_  X  ->  k  C_  X ) )
21 sseq1 3212 . . . . . . . . . . . 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 3371 . . . . . . . . . . 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 2357 . . . . . . . . . . . 12  |-  ( k  =  A  ->  (
x  e.  k  <->  x  e.  A ) )
3029biimprd 214 . . . . . . . . . . 11  |-  ( k  =  A  ->  (
x  e.  A  ->  x  e.  k )
)
31 eleq2 2357 . . . . . . . . . . . 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 5882 . . . . . . . . . . . . 13  |-  ( k  =  A  ->  ( Jt  k )  =  ( Jt  A ) )
3837eleq1d 2362 . . . . . . . . . . . 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 5882 . . . . . . . . . . . . 13  |-  ( k  =  B  ->  ( Jt  k )  =  ( Jt  B ) )
4140eleq1d 2362 . . . . . . . . . . . 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 17170 . . . . . . 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 2371 . . . . . 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 1626 . . 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 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {cpr 3654   U.cuni 3843   U_ciun 3921   ` cfv 5271  (class class class)co 5874   ↾t crest 13341  TopOnctopon 16648   Conccon 17153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-en 6880  df-fin 6883  df-fi 7181  df-rest 13343  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-con 17154
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