isfil2 - Metamath Proof Explorer

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Theorem isfil2 21660

Description: Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

**Assertion**
Ref	Expression
isfil2	$/\$ $/\$ $/\$ $/\$

Distinct variable groups:

Proof of Theorem isfil2 Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		filsspw 21655	. . . 4
2		0nelfil 21653	. . . 4
3		filtop 21659	. . . 4
4	1, 2, 3	3jca 1242	. . 3 $/\$ $/\$
5		elpwi 4168	. . . . 5
6		filss 21657	. . . . . . . . 9 $/\$ $/\$ $/\$
7	6	3exp2 1285	. . . . . . . 8
8	7	com23 86	. . . . . . 7
9	8	imp 445	. . . . . 6 $/\$
10	9	rexlimdv 3030	. . . . 5 $/\$
11	5, 10	sylan2 491	. . . 4 $/\$
12	11	ralrimiva 2966	. . 3
13		filin 21658	. . . . 5 $/\$ $/\$
14	13	3expb 1266	. . . 4 $/\$ $/\$
15	14	ralrimivva 2971	. . 3
16	4, 12, 15	3jca 1242	. 2 $/\$ $/\$ $/\$ $/\$
17		simp11 1091	. . . 4 $/\$ $/\$ $/\$ $/\$
18		simp13 1093	. . . . . 6 $/\$ $/\$ $/\$ $/\$
19		ne0i 3921	. . . . . 6
20	18, 19	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$
21		simp12 1092	. . . . . 6 $/\$ $/\$ $/\$ $/\$
22		df-nel 2898	. . . . . 6
23	21, 22	sylibr 224	. . . . 5 $/\$ $/\$ $/\$ $/\$
24		ssid 3624	. . . . . . . . 9
25		sseq1 3626	. . . . . . . . . 10
26	25	rspcev 3309	. . . . . . . . 9 $/\$
27	24, 26	mpan2 707	. . . . . . . 8
28	27	ralimi 2952	. . . . . . 7
29	28	ralimi 2952	. . . . . 6
30	29	3ad2ant3 1084	. . . . 5 $/\$ $/\$ $/\$ $/\$
31	20, 23, 30	3jca 1242	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		isfbas2 21639	. . . . 5 $/\$ $/\$ $/\$
33	18, 32	syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	17, 31, 33	mpbir2and 957	. . 3 $/\$ $/\$ $/\$ $/\$
35		n0 3931	. . . . . . . 8
36		elin 3796	. . . . . . . . . 10 $/\$
37		elpwi 4168	. . . . . . . . . . 11
38	37	anim2i 593	. . . . . . . . . 10 $/\$ $/\$
39	36, 38	sylbi 207	. . . . . . . . 9 $/\$
40	39	eximi 1762	. . . . . . . 8 $/\$
41	35, 40	sylbi 207	. . . . . . 7 $/\$
42		df-rex 2918	. . . . . . 7 $/\$
43	41, 42	sylibr 224	. . . . . 6
44	43	imim1i 63	. . . . 5
45	44	ralimi 2952	. . . 4
46	45	3ad2ant2 1083	. . 3 $/\$ $/\$ $/\$ $/\$
47		isfil 21651	. . 3 $/\$
48	34, 46, 47	sylanbrc 698	. 2 $/\$ $/\$ $/\$ $/\$
49	16, 48	impbii 199	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wex 1704

wcel 1990

wne 2794

wnel 2897

wral 2912

wrex 2913

cin 3573

wss 3574

c0 3915

cpw 4158

cfv 5888

cfbas 19734

cfil 21649

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-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-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-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-fv 5896 df-fbas 19743 df-fil 21650

This theorem is referenced by: isfild 21662 infil 21667 neifil 21684 trfil2 21691

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