Theorem fnessex 32341

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

Step	Hyp	Ref	Expression
1		fnetg 32340	. . 3 |- ( A Fne B -> A C_ ( topGen ` B ) )
2	1	sselda 3603	. 2 |- ( ( A Fne B /\ S e. A ) -> S e. ( topGen ` B ) )
3		tg2 20769	. 2 |- ( ( S e. ( topGen ` B ) /\ P e. S ) -> E. x e. B ( P e. x /\ x C_ S ) )
4	2, 3	stoic3 1701	1 |- ( ( A Fne B /\ S e. A /\ P e. S ) -> E. x e. B ( P e. x /\ x C_ S ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   E.wrex 2913   C_ wss 3574   class class class wbr 4653   ` cfv 5888   topGenctg 16098   Fnecfne 32331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-topgen 16104  df-fne 32332

This theorem is referenced by:  fneint  32343  fnessref  32352