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Mirrors > Home > ILE Home > Th. List > baspartn | Unicode version |
Description: A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
baspartn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . . . . . . 9 | |
2 | pwidg 3524 | . . . . . . . . 9 | |
3 | 1, 2 | elind 3261 | . . . . . . . 8 |
4 | elssuni 3764 | . . . . . . . 8 | |
5 | 3, 4 | syl 14 | . . . . . . 7 |
6 | inidm 3285 | . . . . . . . . 9 | |
7 | ineq2 3271 | . . . . . . . . 9 | |
8 | 6, 7 | syl5eqr 2186 | . . . . . . . 8 |
9 | 8 | pweqd 3515 | . . . . . . . . . 10 |
10 | 9 | ineq2d 3277 | . . . . . . . . 9 |
11 | 10 | unieqd 3747 | . . . . . . . 8 |
12 | 8, 11 | sseq12d 3128 | . . . . . . 7 |
13 | 5, 12 | syl5ibcom 154 | . . . . . 6 |
14 | 0ss 3401 | . . . . . . . 8 | |
15 | sseq1 3120 | . . . . . . . 8 | |
16 | 14, 15 | mpbiri 167 | . . . . . . 7 |
17 | 16 | a1i 9 | . . . . . 6 |
18 | 13, 17 | jaod 706 | . . . . 5 |
19 | 18 | ralimdv 2500 | . . . 4 |
20 | 19 | ralimia 2493 | . . 3 |
21 | 20 | adantl 275 | . 2 |
22 | isbasisg 12211 | . . 3 | |
23 | 22 | adantr 274 | . 2 |
24 | 21, 23 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 wral 2416 cin 3070 wss 3071 c0 3363 cpw 3510 cuni 3736 ctb 12209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-uni 3737 df-bases 12210 |
This theorem is referenced by: (None) |
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