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Theorem cvmsdisj 24914
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 2573 . . 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 24909 . . . . . . . . . 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 971 . . . . . . . . 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 450 . . . . . . . 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 444 . . . . . . . . 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 2745 . . . . . . . 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 3789 . . . . . . . . . 10  |-  ( u  =  A  ->  { u }  =  { A } )
109difeq2d 3429 . . . . . . . . 9  |-  ( u  =  A  ->  ( T  \  { u }
)  =  ( T 
\  { A }
) )
11 ineq1 3499 . . . . . . . . . 10  |-  ( u  =  A  ->  (
u  i^i  v )  =  ( A  i^i  v ) )
1211eqeq1d 2416 . . . . . . . . 9  |-  ( u  =  A  ->  (
( u  i^i  v
)  =  (/)  <->  ( A  i^i  v )  =  (/) ) )
1310, 12raleqbidv 2880 . . . . . . . 8  |-  ( u  =  A  ->  ( A. v  e.  ( T  \  { u }
) ( u  i^i  v )  =  (/)  <->  A. v  e.  ( T  \  { A } ) ( A  i^i  v
)  =  (/) ) )
1413rspccva 3015 . . . . . . 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 458 . . . . . 6  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  A. v  e.  ( T  \  { A } ) ( A  i^i  v )  =  (/) )
16 necom 2652 . . . . . . 7  |-  ( A  =/=  B  <->  B  =/=  A )
17 eldifsn 3891 . . . . . . . 8  |-  ( B  e.  ( T  \  { A } )  <->  ( B  e.  T  /\  B  =/= 
A ) )
1817biimpri 198 . . . . . . 7  |-  ( ( B  e.  T  /\  B  =/=  A )  ->  B  e.  ( T  \  { A } ) )
1916, 18sylan2b 462 . . . . . 6  |-  ( ( B  e.  T  /\  A  =/=  B )  ->  B  e.  ( T  \  { A } ) )
20 ineq2 3500 . . . . . . . 8  |-  ( v  =  B  ->  ( A  i^i  v )  =  ( A  i^i  B
) )
2120eqeq1d 2416 . . . . . . 7  |-  ( v  =  B  ->  (
( A  i^i  v
)  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
2221rspccv 3013 . . . . . 6  |-  ( A. v  e.  ( T  \  { A } ) ( A  i^i  v
)  =  (/)  ->  ( B  e.  ( T  \  { A } )  ->  ( A  i^i  B )  =  (/) ) )
2315, 19, 22syl2im 36 . . . . 5  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( ( B  e.  T  /\  A  =/= 
B )  ->  ( A  i^i  B )  =  (/) ) )
2423exp3a 426 . . . 4  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( B  e.  T  ->  ( A  =/=  B  ->  ( A  i^i  B
)  =  (/) ) ) )
25243impia 1150 . . 3  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T  /\  B  e.  T )  ->  ( A  =/=  B  ->  ( A  i^i  B
)  =  (/) ) )
261, 25syl5bir 210 . 2  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T  /\  B  e.  T )  ->  ( -.  A  =  B  ->  ( A  i^i  B )  =  (/) ) )
2726orrd 368 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 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670   {crab 2674    \ cdif 3281    i^i cin 3283    C_ wss 3284   (/)c0 3592   ~Pcpw 3763   {csn 3778   U.cuni 3979    e. cmpt 4230   `'ccnv 4840    |` cres 4843   "cima 4844   ` cfv 5417  (class class class)co 6044   ↾t crest 13607    Homeo chmeo 17742
This theorem is referenced by:  cvmscld  24917  cvmsss2  24918  cvmseu  24920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047
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