filuni - Metamath Proof Explorer

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Theorem filuni 21689

Description: The union of a nonempty set of filters with a common base and closed under pairwise union is a filter. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

**Assertion**
Ref	Expression
filuni	$/\$ $/\$

Distinct variable groups:

Proof of Theorem filuni Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni2 4440	. . . 4
2		ssel2 3598	. . . . . . 7 $/\$
3		filelss 21656	. . . . . . . 8 $/\$
4	3	ex 450	. . . . . . 7
5	2, 4	syl 17	. . . . . 6 $/\$
6	5	rexlimdva 3031	. . . . 5
7	6	3ad2ant1 1082	. . . 4 $/\$ $/\$
8	1, 7	syl5bi 232	. . 3 $/\$ $/\$
9	8	pm4.71rd 667	. 2 $/\$ $/\$ $/\$
10		ssn0 3976	. . . 4 $/\$
11		fvprc 6185	. . . . 5
12	11	necon1ai 2821	. . . 4
13	10, 12	syl 17	. . 3 $/\$
14	13	3adant3 1081	. 2 $/\$ $/\$
15		filtop 21659	. . . . . . . . 9
16	2, 15	syl 17	. . . . . . . 8 $/\$
17	16	a1d 25	. . . . . . 7 $/\$
18	17	ralimdva 2962	. . . . . 6
19		r19.2z 4060	. . . . . . 7 $/\$
20	19	ex 450	. . . . . 6
21	18, 20	sylan9 689	. . . . 5 $/\$
22	21	3impia 1261	. . . 4 $/\$ $/\$
23		eluni2 4440	. . . 4
24	22, 23	sylibr 224	. . 3 $/\$ $/\$
25		sbcel1v 3495	. . 3
26	24, 25	sylibr 224	. 2 $/\$ $/\$
27		0nelfil 21653	. . . . . 6
28	2, 27	syl 17	. . . . 5 $/\$
29	28	ralrimiva 2966	. . . 4
30	29	3ad2ant1 1082	. . 3 $/\$ $/\$
31		sbcel1v 3495	. . . . . 6
32		eluni2 4440	. . . . . 6
33	31, 32	bitri 264	. . . . 5
34	33	notbii 310	. . . 4
35		ralnex 2992	. . . 4
36	34, 35	bitr4i 267	. . 3
37	30, 36	sylibr 224	. 2 $/\$ $/\$
38		simp13 1093	. . . . 5 $/\$ $/\$ $/\$ $/\$
39		r19.29 3072	. . . . . 6 $/\$ $/\$
40	39	ex 450	. . . . 5 $/\$
41	38, 40	syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
42		simp1 1061	. . . . 5 $/\$ $/\$
43		simp1 1061	. . . . . . . . 9 $/\$ $/\$
44		simpl 473	. . . . . . . . 9 $/\$ $/\$
45	43, 44, 2	syl2an 494	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
46		simprrr 805	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
47		simpl2 1065	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
48		simpl3 1066	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
49		filss 21657	. . . . . . . 8 $/\$ $/\$ $/\$
50	45, 46, 47, 48, 49	syl13anc 1328	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
51	50	expr 643	. . . . . 6 $/\$ $/\$ $/\$ $/\$
52	51	reximdva 3017	. . . . 5 $/\$ $/\$ $/\$
53	42, 52	syl3an1 1359	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
54	41, 53	syld 47	. . 3 $/\$ $/\$ $/\$ $/\$
55		sbcel1v 3495	. . . 4
56	55, 1	bitri 264	. . 3
57		sbcel1v 3495	. . . 4
58		eluni2 4440	. . . 4
59	57, 58	bitri 264	. . 3
60	54, 56, 59	3imtr4g 285	. 2 $/\$ $/\$ $/\$ $/\$
61		simp13 1093	. . . . 5 $/\$ $/\$ $/\$ $/\$
62		r19.29 3072	. . . . . 6 $/\$ $/\$
63	62	ex 450	. . . . 5 $/\$
64	61, 63	syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
65		simp11 1091	. . . . 5 $/\$ $/\$ $/\$ $/\$
66		r19.29 3072	. . . . . . . . . . 11 $/\$ $/\$
67	66	ex 450	. . . . . . . . . 10 $/\$
68		elun1 3780	. . . . . . . . . . . . . . . 16
69		elun2 3781	. . . . . . . . . . . . . . . 16
70	68, 69	anim12i 590	. . . . . . . . . . . . . . 15 $/\$ $/\$
71		eleq2 2690	. . . . . . . . . . . . . . . . 17
72		eleq2 2690	. . . . . . . . . . . . . . . . 17
73	71, 72	anbi12d 747	. . . . . . . . . . . . . . . 16 $/\$ $/\$
74	73	rspcev 3309	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
75	70, 74	sylan2 491	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
76	75	an12s 843	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
77	76	ex 450	. . . . . . . . . . . 12 $/\$ $/\$
78	77	ad2antlr 763	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
79	78	rexlimdva 3031	. . . . . . . . . 10 $/\$ $/\$ $/\$
80	67, 79	syl9r 78	. . . . . . . . 9 $/\$ $/\$
81	80	impr 649	. . . . . . . 8 $/\$ $/\$ $/\$
82	81	ancom2s 844	. . . . . . 7 $/\$ $/\$ $/\$
83	82	rexlimiva 3028	. . . . . 6 $/\$ $/\$
84	83	imp 445	. . . . 5 $/\$ $/\$ $/\$
85		ssel2 3598	. . . . . . 7 $/\$
86		filin 21658	. . . . . . . 8 $/\$ $/\$
87	86	3expib 1268	. . . . . . 7 $/\$
88	85, 87	syl 17	. . . . . 6 $/\$ $/\$
89	88	reximdva 3017	. . . . 5 $/\$
90	65, 84, 89	syl2im 40	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
91	64, 90	syland 498	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
92		eluni2 4440	. . . . 5
93	55, 92	bitri 264	. . . 4
94	59, 93	anbi12i 733	. . 3 $/\$ $/\$
95		sbcel1v 3495	. . . 4
96		eluni2 4440	. . . 4
97	95, 96	bitri 264	. . 3
98	91, 94, 97	3imtr4g 285	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
99	9, 14, 26, 37, 60, 98	isfild 21662	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wral 2912

wrex 2913

cvv 3200

wsbc 3435

cun 3572

cin 3573

wss 3574

c0 3915

cuni 4436

cfv 5888

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: filssufilg 21715

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