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Mirrors > Home > MPE Home > Th. List > baspartn | Structured version Visualization version 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 22 | . . . . . . . . 9 | |
2 | pwidg 4173 | . . . . . . . . 9 | |
3 | 1, 2 | elind 3798 | . . . . . . . 8 |
4 | elssuni 4467 | . . . . . . . 8 | |
5 | 3, 4 | syl 17 | . . . . . . 7 |
6 | inidm 3822 | . . . . . . . . 9 | |
7 | ineq2 3808 | . . . . . . . . 9 | |
8 | 6, 7 | syl5eqr 2670 | . . . . . . . 8 |
9 | 8 | pweqd 4163 | . . . . . . . . . 10 |
10 | 9 | ineq2d 3814 | . . . . . . . . 9 |
11 | 10 | unieqd 4446 | . . . . . . . 8 |
12 | 8, 11 | sseq12d 3634 | . . . . . . 7 |
13 | 5, 12 | syl5ibcom 235 | . . . . . 6 |
14 | 0ss 3972 | . . . . . . . 8 | |
15 | sseq1 3626 | . . . . . . . 8 | |
16 | 14, 15 | mpbiri 248 | . . . . . . 7 |
17 | 16 | a1i 11 | . . . . . 6 |
18 | 13, 17 | jaod 395 | . . . . 5 |
19 | 18 | ralimdv 2963 | . . . 4 |
20 | 19 | ralimia 2950 | . . 3 |
21 | 20 | adantl 482 | . 2 |
22 | isbasisg 20751 | . . 3 | |
23 | 22 | adantr 481 | . 2 |
24 | 21, 23 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 wa 384 wceq 1483 wcel 1990 wral 2912 cin 3573 wss 3574 c0 3915 cpw 4158 cuni 4436 ctb 20749 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-uni 4437 df-bases 20750 |
This theorem is referenced by: kelac2lem 37634 |
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