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Theorem eltg2b 22813
Description: Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
Assertion
Ref Expression
eltg2b (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴)))
Distinct variable groups:   π‘₯,𝑦,𝐴   π‘₯,𝐡,𝑦   π‘₯,𝑉,𝑦

Proof of Theorem eltg2b
StepHypRef Expression
1 eltg2 22812 . 2 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))))
2 simpl 482 . . . . . . 7 ((π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴) β†’ π‘₯ ∈ 𝑦)
32reximi 3078 . . . . . 6 (βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴) β†’ βˆƒπ‘¦ ∈ 𝐡 π‘₯ ∈ 𝑦)
4 eluni2 4906 . . . . . 6 (π‘₯ ∈ βˆͺ 𝐡 ↔ βˆƒπ‘¦ ∈ 𝐡 π‘₯ ∈ 𝑦)
53, 4sylibr 233 . . . . 5 (βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴) β†’ π‘₯ ∈ βˆͺ 𝐡)
65ralimi 3077 . . . 4 (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴) β†’ βˆ€π‘₯ ∈ 𝐴 π‘₯ ∈ βˆͺ 𝐡)
7 dfss3 3965 . . . 4 (𝐴 βŠ† βˆͺ 𝐡 ↔ βˆ€π‘₯ ∈ 𝐴 π‘₯ ∈ βˆͺ 𝐡)
86, 7sylibr 233 . . 3 (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴) β†’ 𝐴 βŠ† βˆͺ 𝐡)
98pm4.71ri 560 . 2 (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴) ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴)))
101, 9bitr4di 289 1 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064   βŠ† wss 3943  βˆͺ cuni 4902  β€˜cfv 6536  topGenctg 17390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-topgen 17396
This theorem is referenced by:  tg2  22819  tgcl  22823  eltop2  22829  tgss2  22841  basgen2  22843  2ndc1stc  23306  eltx  23423  tgqioo  24667  isfne2  35735
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