Theorem neipeltop 20933

Description: Lemma for neiptopreu 20937. (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

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . 4 |- ( a = C -> ( a e. ( N ` p ) <-> C e. ( N ` p ) ) )
2	1	raleqbi1dv 3146	. . 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 ) }
4	2, 3	elrab2 3366	. 2 |- ( C e. J <-> ( C e. ~P X /\ A. p e. C C e. ( N ` p ) ) )
5		0ex 4790	. . . . . . 7 |- (/) e. _V
6		eleq1 2689	. . . . . . 7 |- ( C = (/) -> ( C e. _V <-> (/) e. _V ) )
7	5, 6	mpbiri 248	. . . . . 6 |- ( C = (/) -> C e. _V )
8	7	adantl 482	. . . . 5 |- ( ( A. p e. C C e. ( N ` p ) /\ C = (/) ) -> C e. _V )
9		elex 3212	. . . . . . 7 |- ( C e. ( N ` p ) -> C e. _V )
10	9	ralimi 2952	. . . . . 6 |- ( A. p e. C C e. ( N ` p ) -> A. p e. C C e. _V )
11		r19.3rzv 4064	. . . . . . 7 |- ( C =/= (/) -> ( C e. _V <-> A. p e. C C e. _V ) )
12	11	biimparc 504	. . . . . 6 |- ( ( A. p e. C C e. _V /\ C =/= (/) ) -> C e. _V )
13	10, 12	sylan 488	. . . . 5 |- ( ( A. p e. C C e. ( N ` p ) /\ C =/= (/) ) -> C e. _V )
14	8, 13	pm2.61dane 2881	. . . 4 |- ( A. p e. C C e. ( N ` p ) -> C e. _V )
15		elpwg 4166	. . . 4 |- ( C e. _V -> ( C e. ~P X <-> C C_ X ) )
16	14, 15	syl 17	. . 3 |- ( A. p e. C C e. ( N ` p ) -> ( C e. ~P X <-> C C_ X ) )
17	16	pm5.32ri 670	. 2 |- ( ( C e. ~P X /\ A. p e. C C e. ( N ` p ) ) <-> ( C C_ X /\ A. p e. C C e. ( N ` p ) ) )
18	4, 17	bitri 264	1 |- ( C e. J <-> ( C C_ X /\ A. p e. C C e. ( N ` p ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160

This theorem is referenced by:  neiptopuni  20934  neiptoptop  20935  neiptopnei  20936  neiptopreu  20937