Theorem fneint 32343

Description: If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)

Assertion

Ref	Expression
fneint	|- ( A Fne B -> |^| { x e. B | P e. x } C_ |^| { x e. A | P e. x } )

Distinct variable groups:   x, A   x, B   x, P

Proof of Theorem fneint

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2690	. . . . 5 |- ( x = y -> ( P e. x <-> P e. y ) )
2	1	elrab 3363	. . . 4 |- ( y e. { x e. A | P e. x } <-> ( y e. A /\ P e. y ) )
3		fnessex 32341	. . . . . . 7 |- ( ( A Fne B /\ y e. A /\ P e. y ) -> E. z e. B ( P e. z /\ z C_ y ) )
4	3	3expb 1266	. . . . . 6 |- ( ( A Fne B /\ ( y e. A /\ P e. y ) ) -> E. z e. B ( P e. z /\ z C_ y ) )
5		eleq2 2690	. . . . . . . . . 10 |- ( x = z -> ( P e. x <-> P e. z ) )
6	5	intminss 4503	. . . . . . . . 9 |- ( ( z e. B /\ P e. z ) -> |^| { x e. B | P e. x } C_ z )
7		sstr 3611	. . . . . . . . 9 |- ( ( |^| { x e. B | P e. x } C_ z /\ z C_ y ) -> |^| { x e. B | P e. x } C_ y )
8	6, 7	sylan 488	. . . . . . . 8 |- ( ( ( z e. B /\ P e. z ) /\ z C_ y ) -> |^| { x e. B | P e. x } C_ y )
9	8	expl 648	. . . . . . 7 |- ( z e. B -> ( ( P e. z /\ z C_ y ) -> |^| { x e. B | P e. x } C_ y ) )
10	9	rexlimiv 3027	. . . . . 6 |- ( E. z e. B ( P e. z /\ z C_ y ) -> |^| { x e. B | P e. x } C_ y )
11	4, 10	syl 17	. . . . 5 |- ( ( A Fne B /\ ( y e. A /\ P e. y ) ) -> |^| { x e. B | P e. x } C_ y )
12	11	ex 450	. . . 4 |- ( A Fne B -> ( ( y e. A /\ P e. y ) -> |^| { x e. B | P e. x } C_ y ) )
13	2, 12	syl5bi 232	. . 3 |- ( A Fne B -> ( y e. { x e. A | P e. x } -> |^| { x e. B | P e. x } C_ y ) )
14	13	ralrimiv 2965	. 2 |- ( A Fne B -> A. y e. { x e. A | P e. x } |^| { x e. B | P e. x } C_ y )
15		ssint 4493	. 2 |- ( |^| { x e. B | P e. x } C_ |^| { x e. A | P e. x } <-> A. y e. { x e. A | P e. x } |^| { x e. B | P e. x } C_ y )
16	14, 15	sylibr 224	1 |- ( A Fne B -> |^| { x e. B | P e. x } C_ |^| { x e. A | P e. x } )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   |^|cint 4475   class class class wbr 4653   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-int 4476  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: (None)