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Theorem cvmsdisj 24962
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 2603 . . 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 24957 . . . . . . . . . 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 972 . . . . . . . . 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 451 . . . . . . . 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 445 . . . . . . . . 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 2783 . . . . . . . 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 3827 . . . . . . . . . 10  |-  ( u  =  A  ->  { u }  =  { A } )
109difeq2d 3467 . . . . . . . . 9  |-  ( u  =  A  ->  ( T  \  { u }
)  =  ( T 
\  { A }
) )
11 ineq1 3537 . . . . . . . . . 10  |-  ( u  =  A  ->  (
u  i^i  v )  =  ( A  i^i  v ) )
1211eqeq1d 2446 . . . . . . . . 9  |-  ( u  =  A  ->  (
( u  i^i  v
)  =  (/)  <->  ( A  i^i  v )  =  (/) ) )
1310, 12raleqbidv 2918 . . . . . . . 8  |-  ( u  =  A  ->  ( A. v  e.  ( T  \  { u }
) ( u  i^i  v )  =  (/)  <->  A. v  e.  ( T  \  { A } ) ( A  i^i  v
)  =  (/) ) )
1413rspccva 3053 . . . . . . 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 459 . . . . . 6  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  A. v  e.  ( T  \  { A } ) ( A  i^i  v )  =  (/) )
16 necom 2687 . . . . . . 7  |-  ( A  =/=  B  <->  B  =/=  A )
17 eldifsn 3929 . . . . . . . 8  |-  ( B  e.  ( T  \  { A } )  <->  ( B  e.  T  /\  B  =/= 
A ) )
1817biimpri 199 . . . . . . 7  |-  ( ( B  e.  T  /\  B  =/=  A )  ->  B  e.  ( T  \  { A } ) )
1916, 18sylan2b 463 . . . . . 6  |-  ( ( B  e.  T  /\  A  =/=  B )  ->  B  e.  ( T  \  { A } ) )
20 ineq2 3538 . . . . . . . 8  |-  ( v  =  B  ->  ( A  i^i  v )  =  ( A  i^i  B
) )
2120eqeq1d 2446 . . . . . . 7  |-  ( v  =  B  ->  (
( A  i^i  v
)  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
2221rspccv 3051 . . . . . 6  |-  ( A. v  e.  ( T  \  { A } ) ( A  i^i  v
)  =  (/)  ->  ( B  e.  ( T  \  { A } )  ->  ( A  i^i  B )  =  (/) ) )
2315, 19, 22syl2im 37 . . . . 5  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( ( B  e.  T  /\  A  =/= 
B )  ->  ( A  i^i  B )  =  (/) ) )
2423exp3a 427 . . . 4  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( B  e.  T  ->  ( A  =/=  B  ->  ( A  i^i  B
)  =  (/) ) ) )
25243impia 1151 . . 3  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T  /\  B  e.  T )  ->  ( A  =/=  B  ->  ( A  i^i  B
)  =  (/) ) )
261, 25syl5bir 211 . 2  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T  /\  B  e.  T )  ->  ( -.  A  =  B  ->  ( A  i^i  B )  =  (/) ) )
2726orrd 369 1  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T  /\  B  e.  T )  ->  ( A  =  B  \/  ( A  i^i  B )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   U.cuni 4017    e. cmpt 4269   `'ccnv 4880    |` cres 4883   "cima 4884   ` cfv 5457  (class class class)co 6084   ↾t crest 13653    Homeo chmeo 17790
This theorem is referenced by:  cvmscld  24965  cvmsss2  24966  cvmseu  24968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087
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