Theorem hausflimlem 21783

Description: If A and B are both limits of the same filter, then all neighborhoods of A and B intersect. (Contributed by Mario Carneiro, 21-Sep-2015.)

Assertion

Ref	Expression
hausflimlem	|- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> ( U i^i V ) =/= (/) )

Proof of Theorem hausflimlem

Step	Hyp	Ref	Expression
1		simp1l 1085	. . 3 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> A e. ( J fLim F ) )
2		eqid 2622	. . . 4 |- U. J = U. J
3	2	flimfil 21773	. . 3 |- ( A e. ( J fLim F ) -> F e. ( Fil ` U. J ) )
4	1, 3	syl 17	. 2 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> F e. ( Fil ` U. J ) )
5		flimtop 21769	. . . . 5 |- ( A e. ( J fLim F ) -> J e. Top )
6	1, 5	syl 17	. . . 4 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> J e. Top )
7		simp2l 1087	. . . 4 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> U e. J )
8		simp3l 1089	. . . 4 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> A e. U )
9		opnneip 20923	. . . 4 |- ( ( J e. Top /\ U e. J /\ A e. U ) -> U e. ( ( nei ` J ) ` { A } ) )
10	6, 7, 8, 9	syl3anc 1326	. . 3 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> U e. ( ( nei ` J ) ` { A } ) )
11		flimnei 21771	. . 3 |- ( ( A e. ( J fLim F ) /\ U e. ( ( nei ` J ) ` { A } ) ) -> U e. F )
12	1, 10, 11	syl2anc 693	. 2 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> U e. F )
13		simp1r 1086	. . 3 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> B e. ( J fLim F ) )
14		simp2r 1088	. . . 4 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> V e. J )
15		simp3r 1090	. . . 4 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> B e. V )
16		opnneip 20923	. . . 4 |- ( ( J e. Top /\ V e. J /\ B e. V ) -> V e. ( ( nei ` J ) ` { B } ) )
17	6, 14, 15, 16	syl3anc 1326	. . 3 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> V e. ( ( nei ` J ) ` { B } ) )
18		flimnei 21771	. . 3 |- ( ( B e. ( J fLim F ) /\ V e. ( ( nei ` J ) ` { B } ) ) -> V e. F )
19	13, 17, 18	syl2anc 693	. 2 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> V e. F )
20		filinn0 21664	. 2 |- ( ( F e. ( Fil ` U. J ) /\ U e. F /\ V e. F ) -> ( U i^i V ) =/= (/) )
21	4, 12, 19, 20	syl3anc 1326	1 |- ( ( ( A e. ( J fLim F ) /\ B e. ( J fLim F ) ) /\ ( U e. J /\ V e. J ) /\ ( A e. U /\ B e. V ) ) -> ( U i^i V ) =/= (/) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   i^i cin 3573   (/)c0 3915   {csn 4177   U.cuni 4436   ` cfv 5888  (class class class)co 6650   Topctop 20698   neicnei 20901   Filcfil 21649   fLim cflim 21738

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-nel 2898  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-top 20699  df-nei 20902  df-fil 21650  df-flim 21743

This theorem is referenced by:  hausflimi  21784