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Theorem bastop1 16563
 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 16538 . . . . 5
2 tgtop 16543 . . . . . 6
32adantr 453 . . . . 5
41, 3sseqtrd 3135 . . . 4
5 eqss 3115 . . . . 5
65baib 876 . . . 4
74, 6syl 17 . . 3
8 dfss3 3093 . . 3
97, 8syl6bb 254 . 2
10 ssexg 4057 . . . . 5
1110ancoms 441 . . . 4
12 eltg3 16532 . . . 4
1311, 12syl 17 . . 3
1413ralbidv 2527 . 2
159, 14bitrd 246 1
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   wa 360  wex 1537   wceq 1619   wcel 1621  wral 2509  cvv 2727   wss 3078  cuni 3727  cfv 4592  ctg 13216  ctop 16463 This theorem is referenced by:  bastop2  16564 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-topgen 13218  df-top 16468
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