Theorem fness 32344

Description: A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)

Hypotheses

Ref	Expression
fness.1	|- X = U. A
fness.2	|- Y = U. B

Assertion

Ref	Expression
fness	|- ( ( B e. C /\ A C_ B /\ X = Y ) -> A Fne B )

Proof of Theorem fness

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

Step	Hyp	Ref	Expression
1		simp3 1063	. 2 |- ( ( B e. C /\ A C_ B /\ X = Y ) -> X = Y )
2		ssel2 3598	. . . . . . 7 |- ( ( A C_ B /\ x e. A ) -> x e. B )
3	2	3adant3 1081	. . . . . 6 |- ( ( A C_ B /\ x e. A /\ y e. x ) -> x e. B )
4		simp3 1063	. . . . . . 7 |- ( ( A C_ B /\ x e. A /\ y e. x ) -> y e. x )
5		ssid 3624	. . . . . . 7 |- x C_ x
6	4, 5	jctir 561	. . . . . 6 |- ( ( A C_ B /\ x e. A /\ y e. x ) -> ( y e. x /\ x C_ x ) )
7		elequ2 2004	. . . . . . . 8 |- ( z = x -> ( y e. z <-> y e. x ) )
8		sseq1 3626	. . . . . . . 8 |- ( z = x -> ( z C_ x <-> x C_ x ) )
9	7, 8	anbi12d 747	. . . . . . 7 |- ( z = x -> ( ( y e. z /\ z C_ x ) <-> ( y e. x /\ x C_ x ) ) )
10	9	rspcev 3309	. . . . . 6 |- ( ( x e. B /\ ( y e. x /\ x C_ x ) ) -> E. z e. B ( y e. z /\ z C_ x ) )
11	3, 6, 10	syl2anc 693	. . . . 5 |- ( ( A C_ B /\ x e. A /\ y e. x ) -> E. z e. B ( y e. z /\ z C_ x ) )
12	11	3expib 1268	. . . 4 |- ( A C_ B -> ( ( x e. A /\ y e. x ) -> E. z e. B ( y e. z /\ z C_ x ) ) )
13	12	ralrimivv 2970	. . 3 |- ( A C_ B -> A. x e. A A. y e. x E. z e. B ( y e. z /\ z C_ x ) )
14	13	3ad2ant2 1083	. 2 |- ( ( B e. C /\ A C_ B /\ X = Y ) -> A. x e. A A. y e. x E. z e. B ( y e. z /\ z C_ x ) )
15		fness.1	. . . 4 |- X = U. A
16		fness.2	. . . 4 |- Y = U. B
17	15, 16	isfne2 32337	. . 3 |- ( B e. C -> ( A Fne B <-> ( X = Y /\ A. x e. A A. y e. x E. z e. B ( y e. z /\ z C_ x ) ) ) )
18	17	3ad2ant1 1082	. 2 |- ( ( B e. C /\ A C_ B /\ X = Y ) -> ( A Fne B <-> ( X = Y /\ A. x e. A A. y e. x E. z e. B ( y e. z /\ z C_ x ) ) ) )
19	1, 14, 18	mpbir2and 957	1 |- ( ( B e. C /\ A C_ B /\ X = Y ) -> A Fne B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   U.cuni 4436   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-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:  fnessref  32352  refssfne  32353