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Theorem fnessex 33248
Description: If 𝐵 is finer than 𝐴 and 𝑆 is an element of 𝐴, every point in 𝑆 is an element of a subset of 𝑆 which is in 𝐵. (Contributed by Jeff Hankins, 28-Sep-2009.)
Assertion
Ref Expression
fnessex ((𝐴Fne𝐵𝑆𝐴𝑃𝑆) → ∃𝑥𝐵 (𝑃𝑥𝑥𝑆))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑃   𝑥,𝑆

Proof of Theorem fnessex
StepHypRef Expression
1 fnetg 33247 . . 3 (𝐴Fne𝐵𝐴 ⊆ (topGen‘𝐵))
21sselda 3853 . 2 ((𝐴Fne𝐵𝑆𝐴) → 𝑆 ∈ (topGen‘𝐵))
3 tg2 21293 . 2 ((𝑆 ∈ (topGen‘𝐵) ∧ 𝑃𝑆) → ∃𝑥𝐵 (𝑃𝑥𝑥𝑆))
42, 3stoic3 1740 1 ((𝐴Fne𝐵𝑆𝐴𝑃𝑆) → ∃𝑥𝐵 (𝑃𝑥𝑥𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1069  wcel 2051  wrex 3084  wss 3824   class class class wbr 4926  cfv 6186  topGenctg 16566  Fnecfne 33238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-iota 6150  df-fun 6188  df-fv 6194  df-topgen 16572  df-fne 33239
This theorem is referenced by:  fneint  33250  fnessref  33259
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