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Theorem cvmsdisj 27157
Description: An even covering of  U is a disjoint union. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypothesis
Ref Expression
cvmcov.1  |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/)
} )  |  ( U. s  =  ( `' F " k )  /\  A. u  e.  s  ( A. v  e.  ( s  \  {
u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )
Homeo ( Jt  k ) ) ) ) } )
Assertion
Ref Expression
cvmsdisj  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T  /\  B  e.  T )  ->  ( A  =  B  \/  ( A  i^i  B )  =  (/) ) )
Distinct variable groups:    k, s, u, v, C    k, F, s, u, v    k, J, s, u, v    U, k, s, u, v    T, s, u, v    u, A, v    v, B
Allowed substitution hints:    A( k, s)    B( u, k, s)    S( v, u, k, s)    T( k)

Proof of Theorem cvmsdisj
StepHypRef Expression
1 df-ne 2606 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
2 cvmcov.1 . . . . . . . . . . 11  |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/)
} )  |  ( U. s  =  ( `' F " k )  /\  A. u  e.  s  ( A. v  e.  ( s  \  {
u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )
Homeo ( Jt  k ) ) ) ) } )
32cvmsi 27152 . . . . . . . . . 10  |-  ( T  e.  ( S `  U )  ->  ( U  e.  J  /\  ( T  C_  C  /\  T  =/=  (/) )  /\  ( U. T  =  ( `' F " U )  /\  A. u  e.  T  ( A. v  e.  ( T  \  {
u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )
Homeo ( Jt  U ) ) ) ) ) )
43simp3d 1002 . . . . . . . . 9  |-  ( T  e.  ( S `  U )  ->  ( U. T  =  ( `' F " U )  /\  A. u  e.  T  ( A. v  e.  ( T  \  {
u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )
Homeo ( Jt  U ) ) ) ) )
54simprd 463 . . . . . . . 8  |-  ( T  e.  ( S `  U )  ->  A. u  e.  T  ( A. v  e.  ( T  \  { u } ) ( u  i^i  v
)  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )
Homeo ( Jt  U ) ) ) )
6 simpl 457 . . . . . . . . 9  |-  ( ( A. v  e.  ( T  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u ) Homeo ( Jt  U ) ) )  ->  A. v  e.  ( T  \  { u }
) ( u  i^i  v )  =  (/) )
76ralimi 2789 . . . . . . . 8  |-  ( A. u  e.  T  ( A. v  e.  ( T  \  { u }
) ( u  i^i  v )  =  (/)  /\  ( F  |`  u
)  e.  ( ( Ct  u ) Homeo ( Jt  U ) ) )  ->  A. u  e.  T  A. v  e.  ( T  \  { u }
) ( u  i^i  v )  =  (/) )
85, 7syl 16 . . . . . . 7  |-  ( T  e.  ( S `  U )  ->  A. u  e.  T  A. v  e.  ( T  \  {
u } ) ( u  i^i  v )  =  (/) )
9 sneq 3885 . . . . . . . . . 10  |-  ( u  =  A  ->  { u }  =  { A } )
109difeq2d 3472 . . . . . . . . 9  |-  ( u  =  A  ->  ( T  \  { u }
)  =  ( T 
\  { A }
) )
11 ineq1 3543 . . . . . . . . . 10  |-  ( u  =  A  ->  (
u  i^i  v )  =  ( A  i^i  v ) )
1211eqeq1d 2449 . . . . . . . . 9  |-  ( u  =  A  ->  (
( u  i^i  v
)  =  (/)  <->  ( A  i^i  v )  =  (/) ) )
1310, 12raleqbidv 2929 . . . . . . . 8  |-  ( u  =  A  ->  ( A. v  e.  ( T  \  { u }
) ( u  i^i  v )  =  (/)  <->  A. v  e.  ( T  \  { A } ) ( A  i^i  v
)  =  (/) ) )
1413rspccva 3070 . . . . . . 7  |-  ( ( A. u  e.  T  A. v  e.  ( T  \  { u }
) ( u  i^i  v )  =  (/)  /\  A  e.  T )  ->  A. v  e.  ( T  \  { A } ) ( A  i^i  v )  =  (/) )
158, 14sylan 471 . . . . . 6  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  A. v  e.  ( T  \  { A } ) ( A  i^i  v )  =  (/) )
16 necom 2691 . . . . . . 7  |-  ( A  =/=  B  <->  B  =/=  A )
17 eldifsn 3998 . . . . . . . 8  |-  ( B  e.  ( T  \  { A } )  <->  ( B  e.  T  /\  B  =/= 
A ) )
1817biimpri 206 . . . . . . 7  |-  ( ( B  e.  T  /\  B  =/=  A )  ->  B  e.  ( T  \  { A } ) )
1916, 18sylan2b 475 . . . . . 6  |-  ( ( B  e.  T  /\  A  =/=  B )  ->  B  e.  ( T  \  { A } ) )
20 ineq2 3544 . . . . . . . 8  |-  ( v  =  B  ->  ( A  i^i  v )  =  ( A  i^i  B
) )
2120eqeq1d 2449 . . . . . . 7  |-  ( v  =  B  ->  (
( A  i^i  v
)  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
2221rspccv 3068 . . . . . 6  |-  ( A. v  e.  ( T  \  { A } ) ( A  i^i  v
)  =  (/)  ->  ( B  e.  ( T  \  { A } )  ->  ( A  i^i  B )  =  (/) ) )
2315, 19, 22syl2im 38 . . . . 5  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( ( B  e.  T  /\  A  =/= 
B )  ->  ( A  i^i  B )  =  (/) ) )
2423expd 436 . . . 4  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( B  e.  T  ->  ( A  =/=  B  ->  ( A  i^i  B
)  =  (/) ) ) )
25243impia 1184 . . 3  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T  /\  B  e.  T )  ->  ( A  =/=  B  ->  ( A  i^i  B
)  =  (/) ) )
261, 25syl5bir 218 . 2  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T  /\  B  e.  T )  ->  ( -.  A  =  B  ->  ( A  i^i  B )  =  (/) ) )
2726orrd 378 1  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T  /\  B  e.  T )  ->  ( A  =  B  \/  ( A  i^i  B )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   A.wral 2713   {crab 2717    \ cdif 3323    i^i cin 3325    C_ wss 3326   (/)c0 3635   ~Pcpw 3858   {csn 3875   U.cuni 4089    e. cmpt 4348   `'ccnv 4837    |` cres 4840   "cima 4841   ` cfv 5416  (class class class)co 6089   ↾t crest 14357   Homeochmeo 19324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092
This theorem is referenced by:  cvmscld  27160  cvmsss2  27161  cvmseu  27163
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