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Theorem neipeltop 20087
Description: Lemma for neiptopreu 20091. (Contributed by Thierry Arnoux, 6-Jan-2018.)
Hypothesis
Ref Expression
neiptop.o  |-  J  =  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p ) }
Assertion
Ref Expression
neipeltop  |-  ( C  e.  J  <->  ( C  C_  X  /\  A. p  e.  C  C  e.  ( N `  p ) ) )
Distinct variable groups:    p, a, C    N, a    X, a
Allowed substitution hints:    J( p, a)    N( p)    X( p)

Proof of Theorem neipeltop
StepHypRef Expression
1 eleq1 2494 . . . 4  |-  ( a  =  C  ->  (
a  e.  ( N `
 p )  <->  C  e.  ( N `  p ) ) )
21raleqbi1dv 2972 . . 3  |-  ( a  =  C  ->  ( A. p  e.  a 
a  e.  ( N `
 p )  <->  A. p  e.  C  C  e.  ( N `  p ) ) )
3 neiptop.o . . 3  |-  J  =  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p ) }
42, 3elrab2 3173 . 2  |-  ( C  e.  J  <->  ( C  e.  ~P X  /\  A. p  e.  C  C  e.  ( N `  p
) ) )
5 0ex 4499 . . . . . . 7  |-  (/)  e.  _V
6 eleq1 2494 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  e.  _V  <->  (/)  e.  _V ) )
75, 6mpbiri 236 . . . . . 6  |-  ( C  =  (/)  ->  C  e. 
_V )
87adantl 467 . . . . 5  |-  ( ( A. p  e.  C  C  e.  ( N `  p )  /\  C  =  (/) )  ->  C  e.  _V )
9 elex 3031 . . . . . . 7  |-  ( C  e.  ( N `  p )  ->  C  e.  _V )
109ralimi 2758 . . . . . 6  |-  ( A. p  e.  C  C  e.  ( N `  p
)  ->  A. p  e.  C  C  e.  _V )
11 r19.3rzv 3835 . . . . . . 7  |-  ( C  =/=  (/)  ->  ( C  e.  _V  <->  A. p  e.  C  C  e.  _V )
)
1211biimparc 489 . . . . . 6  |-  ( ( A. p  e.  C  C  e.  _V  /\  C  =/=  (/) )  ->  C  e.  _V )
1310, 12sylan 473 . . . . 5  |-  ( ( A. p  e.  C  C  e.  ( N `  p )  /\  C  =/=  (/) )  ->  C  e.  _V )
148, 13pm2.61dane 2688 . . . 4  |-  ( A. p  e.  C  C  e.  ( N `  p
)  ->  C  e.  _V )
15 elpwg 3932 . . . 4  |-  ( C  e.  _V  ->  ( C  e.  ~P X  <->  C 
C_  X ) )
1614, 15syl 17 . . 3  |-  ( A. p  e.  C  C  e.  ( N `  p
)  ->  ( C  e.  ~P X  <->  C  C_  X
) )
1716pm5.32ri 642 . 2  |-  ( ( C  e.  ~P X  /\  A. p  e.  C  C  e.  ( N `  p ) )  <->  ( C  C_  X  /\  A. p  e.  C  C  e.  ( N `  p ) ) )
184, 17bitri 252 1  |-  ( C  e.  J  <->  ( C  C_  X  /\  A. p  e.  C  C  e.  ( N `  p ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   {crab 2718   _Vcvv 3022    C_ wss 3379   (/)c0 3704   ~Pcpw 3924   ` cfv 5544
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-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-nul 4498
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rab 2723  df-v 3024  df-dif 3382  df-in 3386  df-ss 3393  df-nul 3705  df-pw 3926
This theorem is referenced by:  neiptopuni  20088  neiptoptop  20089  neiptopnei  20090  neiptopreu  20091
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