Theorem gneispb 38429

Description: Given a neighborhood N of P, each subset of the neighborhood space containing this neighborhood is also a neighborhood of P. Axiom B of Seifert And Threlfall. (Contributed by RP, 5-Apr-2021.)

Hypothesis

Ref	Expression
gneispace.x	|- X = U. J

Assertion

Ref	Expression
gneispb	|- ( ( J e. Top /\ P e. X /\ N e. ( ( nei ` J ) ` { P } ) ) -> A. s e. ~P X ( N C_ s -> s e. ( ( nei ` J ) ` { P } ) ) )

Distinct variable groups:   J, s   N, s   P, s   X, s

Proof of Theorem gneispb

Step	Hyp	Ref	Expression
1		3simpb 1059	. . . . 5 |- ( ( J e. Top /\ P e. X /\ N e. ( ( nei ` J ) ` { P } ) ) -> ( J e. Top /\ N e. ( ( nei ` J ) ` { P } ) ) )
2	1	ad2antrr 762	. . . 4 |- ( ( ( ( J e. Top /\ P e. X /\ N e. ( ( nei ` J ) ` { P } ) ) /\ s e. ~P X ) /\ N C_ s ) -> ( J e. Top /\ N e. ( ( nei ` J ) ` { P } ) ) )
3		simpr 477	. . . 4 |- ( ( ( ( J e. Top /\ P e. X /\ N e. ( ( nei ` J ) ` { P } ) ) /\ s e. ~P X ) /\ N C_ s ) -> N C_ s )
4		simplr 792	. . . . 5 |- ( ( ( ( J e. Top /\ P e. X /\ N e. ( ( nei ` J ) ` { P } ) ) /\ s e. ~P X ) /\ N C_ s ) -> s e. ~P X )
5	4	elpwid 4170	. . . 4 |- ( ( ( ( J e. Top /\ P e. X /\ N e. ( ( nei ` J ) ` { P } ) ) /\ s e. ~P X ) /\ N C_ s ) -> s C_ X )
6		gneispace.x	. . . . 5 |- X = U. J
7	6	ssnei2 20920	. . . 4 |- ( ( ( J e. Top /\ N e. ( ( nei ` J ) ` { P } ) ) /\ ( N C_ s /\ s C_ X ) ) -> s e. ( ( nei ` J ) ` { P } ) )
8	2, 3, 5, 7	syl12anc 1324	. . 3 |- ( ( ( ( J e. Top /\ P e. X /\ N e. ( ( nei ` J ) ` { P } ) ) /\ s e. ~P X ) /\ N C_ s ) -> s e. ( ( nei ` J ) ` { P } ) )
9	8	exp31 630	. 2 |- ( ( J e. Top /\ P e. X /\ N e. ( ( nei ` J ) ` { P } ) ) -> ( s e. ~P X -> ( N C_ s -> s e. ( ( nei ` J ) ` { P } ) ) ) )
10	9	ralrimiv 2965	1 |- ( ( J e. Top /\ P e. X /\ N e. ( ( nei ` J ) ` { P } ) ) -> A. s e. ~P X ( N C_ s -> s e. ( ( nei ` J ) ` { P } ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   ` cfv 5888   Topctop 20698   neicnei 20901

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-nei 20902

This theorem is referenced by: (None)