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Theorem cvmsdisj 24525
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 2531 . . 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 24520 . . . . . . . . . 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 970 . . . . . . . . 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 449 . . . . . . . 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 443 . . . . . . . . 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 2703 . . . . . . . 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 15 . . . . . . 7  |-  ( T  e.  ( S `  U )  ->  A. u  e.  T  A. v  e.  ( T  \  {
u } ) ( u  i^i  v )  =  (/) )
9 sneq 3740 . . . . . . . . . 10  |-  ( u  =  A  ->  { u }  =  { A } )
109difeq2d 3381 . . . . . . . . 9  |-  ( u  =  A  ->  ( T  \  { u }
)  =  ( T 
\  { A }
) )
11 ineq1 3451 . . . . . . . . . 10  |-  ( u  =  A  ->  (
u  i^i  v )  =  ( A  i^i  v ) )
1211eqeq1d 2374 . . . . . . . . 9  |-  ( u  =  A  ->  (
( u  i^i  v
)  =  (/)  <->  ( A  i^i  v )  =  (/) ) )
1310, 12raleqbidv 2833 . . . . . . . 8  |-  ( u  =  A  ->  ( A. v  e.  ( T  \  { u }
) ( u  i^i  v )  =  (/)  <->  A. v  e.  ( T  \  { A } ) ( A  i^i  v
)  =  (/) ) )
1413rspccva 2968 . . . . . . 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 457 . . . . . 6  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  A. v  e.  ( T  \  { A } ) ( A  i^i  v )  =  (/) )
16 necom 2610 . . . . . . 7  |-  ( A  =/=  B  <->  B  =/=  A )
17 eldifsn 3842 . . . . . . . 8  |-  ( B  e.  ( T  \  { A } )  <->  ( B  e.  T  /\  B  =/= 
A ) )
1817biimpri 197 . . . . . . 7  |-  ( ( B  e.  T  /\  B  =/=  A )  ->  B  e.  ( T  \  { A } ) )
1916, 18sylan2b 461 . . . . . 6  |-  ( ( B  e.  T  /\  A  =/=  B )  ->  B  e.  ( T  \  { A } ) )
20 ineq2 3452 . . . . . . . 8  |-  ( v  =  B  ->  ( A  i^i  v )  =  ( A  i^i  B
) )
2120eqeq1d 2374 . . . . . . 7  |-  ( v  =  B  ->  (
( A  i^i  v
)  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
2221rspccv 2966 . . . . . 6  |-  ( A. v  e.  ( T  \  { A } ) ( A  i^i  v
)  =  (/)  ->  ( B  e.  ( T  \  { A } )  ->  ( A  i^i  B )  =  (/) ) )
2315, 19, 22syl2im 34 . . . . 5  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( ( B  e.  T  /\  A  =/= 
B )  ->  ( A  i^i  B )  =  (/) ) )
2423exp3a 425 . . . 4  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( B  e.  T  ->  ( A  =/=  B  ->  ( A  i^i  B
)  =  (/) ) ) )
25243impia 1149 . . 3  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T  /\  B  e.  T )  ->  ( A  =/=  B  ->  ( A  i^i  B
)  =  (/) ) )
261, 25syl5bir 209 . 2  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T  /\  B  e.  T )  ->  ( -.  A  =  B  ->  ( A  i^i  B )  =  (/) ) )
2726orrd 367 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 357    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   {crab 2632    \ cdif 3235    i^i cin 3237    C_ wss 3238   (/)c0 3543   ~Pcpw 3714   {csn 3729   U.cuni 3929    e. cmpt 4179   `'ccnv 4791    |` cres 4794   "cima 4795   ` cfv 5358  (class class class)co 5981   ↾t crest 13535    Homeo chmeo 17661
This theorem is referenced by:  cvmscld  24528  cvmsss2  24529  cvmseu  24531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984
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