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Theorem bastop1 17050
 Description: A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom " " to express " is a basis for topology ," since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
bastop1
Distinct variable groups:   ,,   ,,

Proof of Theorem bastop1
StepHypRef Expression
1 tgss 17025 . . . . 5
2 tgtop 17030 . . . . . 6
32adantr 452 . . . . 5
41, 3sseqtrd 3376 . . . 4
5 eqss 3355 . . . . 5
65baib 872 . . . 4
74, 6syl 16 . . 3
8 dfss3 3330 . . 3
97, 8syl6bb 253 . 2
10 ssexg 4341 . . . . 5
1110ancoms 440 . . . 4
12 eltg3 17019 . . . 4
1311, 12syl 16 . . 3
1413ralbidv 2717 . 2
159, 14bitrd 245 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1550   wceq 1652   wcel 1725  wral 2697  cvv 2948   wss 3312  cuni 4007  cfv 5446  ctg 13657  ctop 16950 This theorem is referenced by:  bastop2  17051 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-topgen 13659  df-top 16955
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