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Theorem fnessex 35863
Description: If 𝐡 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 𝐡. (Contributed by Jeff Hankins, 28-Sep-2009.)
Assertion
Ref Expression
fnessex ((𝐴Fne𝐡 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝑆) β†’ βˆƒπ‘₯ ∈ 𝐡 (𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑆))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝑃   π‘₯,𝑆

Proof of Theorem fnessex
StepHypRef Expression
1 fnetg 35862 . . 3 (𝐴Fne𝐡 β†’ 𝐴 βŠ† (topGenβ€˜π΅))
21sselda 3982 . 2 ((𝐴Fne𝐡 ∧ 𝑆 ∈ 𝐴) β†’ 𝑆 ∈ (topGenβ€˜π΅))
3 tg2 22888 . 2 ((𝑆 ∈ (topGenβ€˜π΅) ∧ 𝑃 ∈ 𝑆) β†’ βˆƒπ‘₯ ∈ 𝐡 (𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑆))
42, 3stoic3 1770 1 ((𝐴Fne𝐡 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝑆) β†’ βˆƒπ‘₯ ∈ 𝐡 (𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑆))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   ∈ wcel 2098  βˆƒwrex 3067   βŠ† wss 3949   class class class wbr 5152  β€˜cfv 6553  topGenctg 17426  Fnecfne 35853
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-topgen 17432  df-fne 35854
This theorem is referenced by:  fneint  35865  fnessref  35874
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