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Theorem fnessex 32639
 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 32638 . . 3 (𝐴Fne𝐵𝐴 ⊆ (topGen‘𝐵))
21sselda 3736 . 2 ((𝐴Fne𝐵𝑆𝐴) → 𝑆 ∈ (topGen‘𝐵))
3 tg2 20963 . 2 ((𝑆 ∈ (topGen‘𝐵) ∧ 𝑃𝑆) → ∃𝑥𝐵 (𝑃𝑥𝑥𝑆))
42, 3stoic3 1842 1 ((𝐴Fne𝐵𝑆𝐴𝑃𝑆) → ∃𝑥𝐵 (𝑃𝑥𝑥𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   ∈ wcel 2131  ∃wrex 3043   ⊆ wss 3707   class class class wbr 4796  ‘cfv 6041  topGenctg 16292  Fnecfne 32629 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-topgen 16298  df-fne 32630 This theorem is referenced by:  fneint  32641  fnessref  32650
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