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Theorem baspartn 20046

Description: A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

**Assertion**
Ref	Expression
baspartn	$/\$ $\/$

(

)

**Proof of Theorem baspartn**
Step	Hyp	Ref	Expression
1		id 22	. . . . . . . . 9
2		pwidg 3955	. . . . . . . . 9
3	1, 2	elind 3609	. . . . . . . 8
4		elssuni 4219	. . . . . . . 8
5	3, 4	syl 17	. . . . . . 7
6		inidm 3632	. . . . . . . . 9
7		ineq2 3619	. . . . . . . . 9
8	6, 7	syl5eqr 2519	. . . . . . . 8
9	8	pweqd 3947	. . . . . . . . . 10
10	9	ineq2d 3625	. . . . . . . . 9
11	10	unieqd 4200	. . . . . . . 8
12	8, 11	sseq12d 3447	. . . . . . 7
13	5, 12	syl5ibcom 228	. . . . . 6
14		0ss 3766	. . . . . . . 8
15		sseq1 3439	. . . . . . . 8
16	14, 15	mpbiri 241	. . . . . . 7
17	16	a1i 11	. . . . . 6
18	13, 17	jaod 387	. . . . 5 $\/$
19	18	ralimdv 2806	. . . 4 $\/$
20	19	ralimia 2794	. . 3 $\/$
21	20	adantl 473	. 2 $/\$ $\/$
22		isbasisg 20039	. . 3
23	22	adantr 472	. 2 $/\$ $\/$
24	21, 23	mpbird 240	1 $/\$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 189 $\/$ wo 375 $/\$ wa 376

wceq 1452

wcel 1904

wral 2756

cin 3389

wss 3390

c0 3722

cpw 3942

cuni 4190

ctb 19997

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451

This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ral 2761 df-rex 2762 df-v 3033 df-dif 3393 df-in 3397 df-ss 3404 df-nul 3723 df-pw 3944 df-uni 4191 df-bases 19999

This theorem is referenced by: kelac2lem 35993