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Theorem dispcmp 28695
Description: Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
Assertion
Ref Expression
dispcmp  |-  ( X  e.  V  ->  ~P X  e. Paracomp )

Proof of Theorem dispcmp
Dummy variables  v 
y  z  u  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 distop 20010 . . 3  |-  ( X  e.  V  ->  ~P X  e.  Top )
2 simpr 462 . . . . . . . . . . . 12  |-  ( ( x  e.  X  /\  u  =  { x } )  ->  u  =  { x } )
3 snelpwi 4666 . . . . . . . . . . . . 13  |-  ( x  e.  X  ->  { x }  e.  ~P X
)
43adantr 466 . . . . . . . . . . . 12  |-  ( ( x  e.  X  /\  u  =  { x } )  ->  { x }  e.  ~P X
)
52, 4eqeltrd 2507 . . . . . . . . . . 11  |-  ( ( x  e.  X  /\  u  =  { x } )  ->  u  e.  ~P X )
65rexlimiva 2910 . . . . . . . . . 10  |-  ( E. x  e.  X  u  =  { x }  ->  u  e.  ~P X
)
76abssi 3536 . . . . . . . . 9  |-  { u  |  E. x  e.  X  u  =  { x } }  C_  ~P X
8 simpl 458 . . . . . . . . . . . . . 14  |-  ( ( u  =  v  /\  x  =  z )  ->  u  =  v )
9 simpr 462 . . . . . . . . . . . . . . 15  |-  ( ( u  =  v  /\  x  =  z )  ->  x  =  z )
109sneqd 4010 . . . . . . . . . . . . . 14  |-  ( ( u  =  v  /\  x  =  z )  ->  { x }  =  { z } )
118, 10eqeq12d 2444 . . . . . . . . . . . . 13  |-  ( ( u  =  v  /\  x  =  z )  ->  ( u  =  {
x }  <->  v  =  { z } ) )
1211cbvrexdva 3061 . . . . . . . . . . . 12  |-  ( u  =  v  ->  ( E. x  e.  X  u  =  { x } 
<->  E. z  e.  X  v  =  { z } ) )
1312cbvabv 2561 . . . . . . . . . . 11  |-  { u  |  E. x  e.  X  u  =  { x } }  =  {
v  |  E. z  e.  X  v  =  { z } }
1413dissnlocfin 20543 . . . . . . . . . 10  |-  ( X  e.  V  ->  { u  |  E. x  e.  X  u  =  { x } }  e.  ( LocFin `
 ~P X ) )
15 elpwg 3989 . . . . . . . . . 10  |-  ( { u  |  E. x  e.  X  u  =  { x } }  e.  ( LocFin `  ~P X )  ->  ( { u  |  E. x  e.  X  u  =  { x } }  e.  ~P ~P X  <->  { u  |  E. x  e.  X  u  =  { x } }  C_ 
~P X ) )
1614, 15syl 17 . . . . . . . . 9  |-  ( X  e.  V  ->  ( { u  |  E. x  e.  X  u  =  { x } }  e.  ~P ~P X  <->  { u  |  E. x  e.  X  u  =  { x } }  C_  ~P X
) )
177, 16mpbiri 236 . . . . . . . 8  |-  ( X  e.  V  ->  { u  |  E. x  e.  X  u  =  { x } }  e.  ~P ~P X )
1817ad2antrr 730 . . . . . . 7  |-  ( ( ( X  e.  V  /\  y  e.  ~P ~P X )  /\  X  =  U. y )  ->  { u  |  E. x  e.  X  u  =  { x } }  e.  ~P ~P X )
1914ad2antrr 730 . . . . . . 7  |-  ( ( ( X  e.  V  /\  y  e.  ~P ~P X )  /\  X  =  U. y )  ->  { u  |  E. x  e.  X  u  =  { x } }  e.  ( LocFin `  ~P X ) )
2018, 19elind 3650 . . . . . 6  |-  ( ( ( X  e.  V  /\  y  e.  ~P ~P X )  /\  X  =  U. y )  ->  { u  |  E. x  e.  X  u  =  { x } }  e.  ( ~P ~P X  i^i  ( LocFin `  ~P X ) ) )
21 simpll 758 . . . . . . 7  |-  ( ( ( X  e.  V  /\  y  e.  ~P ~P X )  /\  X  =  U. y )  ->  X  e.  V )
22 simpr 462 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  y  e.  ~P ~P X )  /\  X  =  U. y )  ->  X  =  U. y
)
2322eqcomd 2430 . . . . . . 7  |-  ( ( ( X  e.  V  /\  y  e.  ~P ~P X )  /\  X  =  U. y )  ->  U. y  =  X
)
2413dissnref 20542 . . . . . . 7  |-  ( ( X  e.  V  /\  U. y  =  X )  ->  { u  |  E. x  e.  X  u  =  { x } } Ref y )
2521, 23, 24syl2anc 665 . . . . . 6  |-  ( ( ( X  e.  V  /\  y  e.  ~P ~P X )  /\  X  =  U. y )  ->  { u  |  E. x  e.  X  u  =  { x } } Ref y )
26 breq1 4426 . . . . . . 7  |-  ( z  =  { u  |  E. x  e.  X  u  =  { x } }  ->  ( z Ref y  <->  { u  |  E. x  e.  X  u  =  { x } } Ref y ) )
2726rspcev 3182 . . . . . 6  |-  ( ( { u  |  E. x  e.  X  u  =  { x } }  e.  ( ~P ~P X  i^i  ( LocFin `  ~P X ) )  /\  { u  |  E. x  e.  X  u  =  { x } } Ref y )  ->  E. z  e.  ( ~P ~P X  i^i  ( LocFin `  ~P X ) ) z Ref y
)
2820, 25, 27syl2anc 665 . . . . 5  |-  ( ( ( X  e.  V  /\  y  e.  ~P ~P X )  /\  X  =  U. y )  ->  E. z  e.  ( ~P ~P X  i^i  ( LocFin `
 ~P X ) ) z Ref y
)
2928ex 435 . . . 4  |-  ( ( X  e.  V  /\  y  e.  ~P ~P X )  ->  ( X  =  U. y  ->  E. z  e.  ( ~P ~P X  i^i  ( LocFin `  ~P X ) ) z Ref y
) )
3029ralrimiva 2836 . . 3  |-  ( X  e.  V  ->  A. y  e.  ~P  ~P X ( X  =  U. y  ->  E. z  e.  ( ~P ~P X  i^i  ( LocFin `  ~P X ) ) z Ref y
) )
31 unipw 4671 . . . . 5  |-  U. ~P X  =  X
3231eqcomi 2435 . . . 4  |-  X  = 
U. ~P X
3332iscref 28680 . . 3  |-  ( ~P X  e. CovHasRef ( LocFin `  ~P X )  <->  ( ~P X  e.  Top  /\  A. y  e.  ~P  ~P X
( X  =  U. y  ->  E. z  e.  ( ~P ~P X  i^i  ( LocFin `  ~P X ) ) z Ref y
) ) )
341, 30, 33sylanbrc 668 . 2  |-  ( X  e.  V  ->  ~P X  e. CovHasRef ( LocFin `  ~P X ) )
35 ispcmp 28693 . 2  |-  ( ~P X  e. Paracomp  <->  ~P X  e. CovHasRef ( LocFin `  ~P X ) )
3634, 35sylibr 215 1  |-  ( X  e.  V  ->  ~P X  e. Paracomp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2407   A.wral 2771   E.wrex 2772    i^i cin 3435    C_ wss 3436   ~Pcpw 3981   {csn 3998   U.cuni 4219   class class class wbr 4423   ` cfv 5601   Topctop 19916   Refcref 20516   LocFinclocfin 20518  CovHasRefccref 28678  Paracompcpcmp 28691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6708  df-1o 7194  df-en 7582  df-fin 7585  df-top 19920  df-ref 20519  df-locfin 20521  df-cref 28679  df-pcmp 28692
This theorem is referenced by: (None)
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