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Theorem fnessex 26357
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  |-  ( ( A Fne B  /\  S  e.  A  /\  P  e.  S )  ->  E. x  e.  B  ( P  e.  x  /\  x  C_  S ) )
Distinct variable groups:    x, A    x, B    x, P    x, S

Proof of Theorem fnessex
StepHypRef Expression
1 fnetg 26356 . . . 4  |-  ( A Fne B  ->  A  C_  ( topGen `  B )
)
21sselda 3350 . . 3  |-  ( ( A Fne B  /\  S  e.  A )  ->  S  e.  ( topGen `  B ) )
3 tg2 17032 . . 3  |-  ( ( S  e.  ( topGen `  B )  /\  P  e.  S )  ->  E. x  e.  B  ( P  e.  x  /\  x  C_  S ) )
42, 3sylan 459 . 2  |-  ( ( ( A Fne B  /\  S  e.  A
)  /\  P  e.  S )  ->  E. x  e.  B  ( P  e.  x  /\  x  C_  S ) )
543impa 1149 1  |-  ( ( A Fne B  /\  S  e.  A  /\  P  e.  S )  ->  E. x  e.  B  ( P  e.  x  /\  x  C_  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    e. wcel 1726   E.wrex 2708    C_ wss 3322   class class class wbr 4214   ` cfv 5456   topGenctg 13667   Fnecfne 26341
This theorem is referenced by:  fneint  26359  fnessref  26375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-topgen 13669  df-fne 26345
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