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Theorem tg1 22458
Description: Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
Assertion
Ref Expression
tg1 (𝐴 ∈ (topGenβ€˜π΅) β†’ 𝐴 βŠ† βˆͺ 𝐡)

Proof of Theorem tg1
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6925 . 2 (𝐴 ∈ (topGenβ€˜π΅) β†’ 𝐡 ∈ dom topGen)
2 eltg2 22452 . . 3 (𝐡 ∈ dom topGen β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))))
32simprbda 499 . 2 ((𝐡 ∈ dom topGen ∧ 𝐴 ∈ (topGenβ€˜π΅)) β†’ 𝐴 βŠ† βˆͺ 𝐡)
41, 3mpancom 686 1 (𝐴 ∈ (topGenβ€˜π΅) β†’ 𝐴 βŠ† βˆͺ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3947  βˆͺ cuni 4907  dom cdm 5675  β€˜cfv 6540  topGenctg 17379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-topgen 17385
This theorem is referenced by:  unitg  22461  tgcl  22463  ontgval  35304
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