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Theorem neipeltop 19800
Description: Lemma for neiptopreu 19804 (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 2526 . . . 4  |-  ( a  =  C  ->  (
a  e.  ( N `
 p )  <->  C  e.  ( N `  p ) ) )
21raleqbi1dv 3059 . . 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 3256 . 2  |-  ( C  e.  J  <->  ( C  e.  ~P X  /\  A. p  e.  C  C  e.  ( N `  p
) ) )
5 0ex 4569 . . . . . . 7  |-  (/)  e.  _V
6 eleq1 2526 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  e.  _V  <->  (/)  e.  _V ) )
75, 6mpbiri 233 . . . . . 6  |-  ( C  =  (/)  ->  C  e. 
_V )
87adantl 464 . . . . 5  |-  ( ( A. p  e.  C  C  e.  ( N `  p )  /\  C  =  (/) )  ->  C  e.  _V )
9 elex 3115 . . . . . . 7  |-  ( C  e.  ( N `  p )  ->  C  e.  _V )
109ralimi 2847 . . . . . 6  |-  ( A. p  e.  C  C  e.  ( N `  p
)  ->  A. p  e.  C  C  e.  _V )
11 r19.3rzv 3910 . . . . . . 7  |-  ( C  =/=  (/)  ->  ( C  e.  _V  <->  A. p  e.  C  C  e.  _V )
)
1211biimparc 485 . . . . . 6  |-  ( ( A. p  e.  C  C  e.  _V  /\  C  =/=  (/) )  ->  C  e.  _V )
1310, 12sylan 469 . . . . 5  |-  ( ( A. p  e.  C  C  e.  ( N `  p )  /\  C  =/=  (/) )  ->  C  e.  _V )
148, 13pm2.61dane 2772 . . . 4  |-  ( A. p  e.  C  C  e.  ( N `  p
)  ->  C  e.  _V )
15 elpwg 4007 . . . 4  |-  ( C  e.  _V  ->  ( C  e.  ~P X  <->  C 
C_  X ) )
1614, 15syl 16 . . 3  |-  ( A. p  e.  C  C  e.  ( N `  p
)  ->  ( C  e.  ~P X  <->  C  C_  X
) )
1716pm5.32ri 636 . 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 249 1  |-  ( C  e.  J  <->  ( C  C_  X  /\  A. p  e.  C  C  e.  ( N `  p ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {crab 2808   _Vcvv 3106    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-pw 4001
This theorem is referenced by:  neiptopuni  19801  neiptoptop  19802  neiptopnei  19803  neiptopreu  19804
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