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Theorem fnessex 26275
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 26274 . . . 4  |-  ( A Fne B  ->  A  C_  ( topGen `  B )
)
21sselda 3180 . . 3  |-  ( ( A Fne B  /\  S  e.  A )  ->  S  e.  ( topGen `  B ) )
3 tg2 16703 . . 3  |-  ( ( S  e.  ( topGen `  B )  /\  P  e.  S )  ->  E. x  e.  B  ( P  e.  x  /\  x  C_  S ) )
42, 3sylan 457 . 2  |-  ( ( ( A Fne B  /\  S  e.  A
)  /\  P  e.  S )  ->  E. x  e.  B  ( P  e.  x  /\  x  C_  S ) )
543impa 1146 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 358    /\ w3a 934    e. wcel 1684   E.wrex 2544    C_ wss 3152   class class class wbr 4023   ` cfv 5255   topGenctg 13342   Fnecfne 26259
This theorem is referenced by:  fneint  26277  fnessref  26293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-topgen 13344  df-fne 26263
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