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Theorem neipeltop 18849
Description: Lemma for neiptopreu 18853 (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 2523 . . . 4  |-  ( a  =  C  ->  (
a  e.  ( N `
 p )  <->  C  e.  ( N `  p ) ) )
21raleqbi1dv 3021 . . 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 3216 . 2  |-  ( C  e.  J  <->  ( C  e.  ~P X  /\  A. p  e.  C  C  e.  ( N `  p
) ) )
5 0ex 4520 . . . . . . 7  |-  (/)  e.  _V
6 eleq1 2523 . . . . . . 7  |-  ( C  =  (/)  ->  ( C  e.  _V  <->  (/)  e.  _V ) )
75, 6mpbiri 233 . . . . . 6  |-  ( C  =  (/)  ->  C  e. 
_V )
87adantl 466 . . . . 5  |-  ( ( A. p  e.  C  C  e.  ( N `  p )  /\  C  =  (/) )  ->  C  e.  _V )
9 elex 3077 . . . . . . 7  |-  ( C  e.  ( N `  p )  ->  C  e.  _V )
109ralimi 2811 . . . . . 6  |-  ( A. p  e.  C  C  e.  ( N `  p
)  ->  A. p  e.  C  C  e.  _V )
11 r19.3rzv 3871 . . . . . . 7  |-  ( C  =/=  (/)  ->  ( C  e.  _V  <->  A. p  e.  C  C  e.  _V )
)
1211biimparc 487 . . . . . 6  |-  ( ( A. p  e.  C  C  e.  _V  /\  C  =/=  (/) )  ->  C  e.  _V )
1310, 12sylan 471 . . . . 5  |-  ( ( A. p  e.  C  C  e.  ( N `  p )  /\  C  =/=  (/) )  ->  C  e.  _V )
148, 13pm2.61dane 2766 . . . 4  |-  ( A. p  e.  C  C  e.  ( N `  p
)  ->  C  e.  _V )
15 elpwg 3966 . . . 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 638 . 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 369    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   {crab 2799   _Vcvv 3068    C_ wss 3426   (/)c0 3735   ~Pcpw 3958   ` cfv 5516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rab 2804  df-v 3070  df-dif 3429  df-in 3433  df-ss 3440  df-nul 3736  df-pw 3960
This theorem is referenced by:  neiptopuni  18850  neiptoptop  18851  neiptopnei  18852  neiptopreu  18853
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