Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  baspartn Structured version   Visualization version   Unicode version

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
Distinct variable group:   ,,
Allowed substitution hints:   (,)

Proof of Theorem baspartn
StepHypRef Expression
1 id 22 . . . . . . . . 9
2 pwidg 3955 . . . . . . . . 9
31, 2elind 3609 . . . . . . . 8
4 elssuni 4219 . . . . . . . 8
53, 4syl 17 . . . . . . 7
6 inidm 3632 . . . . . . . . 9
7 ineq2 3619 . . . . . . . . 9
86, 7syl5eqr 2519 . . . . . . . 8
98pweqd 3947 . . . . . . . . . 10
109ineq2d 3625 . . . . . . . . 9
1110unieqd 4200 . . . . . . . 8
128, 11sseq12d 3447 . . . . . . 7
135, 12syl5ibcom 228 . . . . . 6
14 0ss 3766 . . . . . . . 8
15 sseq1 3439 . . . . . . . 8
1614, 15mpbiri 241 . . . . . . 7
1716a1i 11 . . . . . 6
1813, 17jaod 387 . . . . 5
1918ralimdv 2806 . . . 4
2019ralimia 2794 . . 3
2120adantl 473 . 2
22 isbasisg 20039 . . 3
2322adantr 472 . 2
2421, 23mpbird 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
 Copyright terms: Public domain W3C validator