Theorem gneispace0nelrn3 38440

Description: A generic neighborhood space has a non-empty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.)

Hypothesis

Ref	Expression
gneispace.a	|- A = { f | ( f : dom f --> ( ~P ( ~P dom f \ { (/) } ) \ { (/) } ) /\ A. p e. dom f A. n e. ( f ` p ) ( p e. n /\ A. s e. ~P dom f ( n C_ s -> s e. ( f ` p ) ) ) ) }

Assertion

Ref	Expression
gneispace0nelrn3	|- ( F e. A -> -. (/) e. ran F )

Distinct variable groups:   n, F, p, f   F, s, f   f, n, p

Allowed substitution hints:   A( f, n, s, p)

Proof of Theorem gneispace0nelrn3

Step	Hyp	Ref	Expression
1		gneispace.a	. . 3 |- A = { f | ( f : dom f --> ( ~P ( ~P dom f \ { (/) } ) \ { (/) } ) /\ A. p e. dom f A. n e. ( f ` p ) ( p e. n /\ A. s e. ~P dom f ( n C_ s -> s e. ( f ` p ) ) ) ) }
2	1	gneispacern 38436	. 2 |- ( F e. A -> ran F C_ ( ~P ( ~P dom F \ { (/) } ) \ { (/) } ) )
3		neldifsnd 4322	. . 3 |- ( ran F C_ ( ~P ( ~P dom F \ { (/) } ) \ { (/) } ) -> -. (/) e. ( ~P ( ~P dom F \ { (/) } ) \ { (/) } ) )
4		ssel 3597	. . 3 |- ( ran F C_ ( ~P ( ~P dom F \ { (/) } ) \ { (/) } ) -> ( (/) e. ran F -> (/) e. ( ~P ( ~P dom F \ { (/) } ) \ { (/) } ) ) )
5	3, 4	mtod 189	. 2 |- ( ran F C_ ( ~P ( ~P dom F \ { (/) } ) \ { (/) } ) -> -. (/) e. ran F )
6	2, 5	syl 17	1 |- ( F e. A -> -. (/) e. ran F )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   dom cdm 5114   ran crn 5115   -->wf 5884   ` 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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by: (None)