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Theorem fness 28697
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.
StepHypRef Expression
1 simp3 990 . 2  |-  ( ( B  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  X  =  Y )
2 ssel2 3454 . . . . . . 7  |-  ( ( A  C_  B  /\  x  e.  A )  ->  x  e.  B )
323adant3 1008 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  A  /\  y  e.  x )  ->  x  e.  B )
4 simp3 990 . . . . . . 7  |-  ( ( A  C_  B  /\  x  e.  A  /\  y  e.  x )  ->  y  e.  x )
5 ssid 3478 . . . . . . 7  |-  x  C_  x
64, 5jctir 538 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  A  /\  y  e.  x )  ->  ( y  e.  x  /\  x  C_  x ) )
7 elequ2 1763 . . . . . . . 8  |-  ( z  =  x  ->  (
y  e.  z  <->  y  e.  x ) )
8 sseq1 3480 . . . . . . . 8  |-  ( z  =  x  ->  (
z  C_  x  <->  x  C_  x
) )
97, 8anbi12d 710 . . . . . . 7  |-  ( z  =  x  ->  (
( y  e.  z  /\  z  C_  x
)  <->  ( y  e.  x  /\  x  C_  x ) ) )
109rspcev 3173 . . . . . 6  |-  ( ( x  e.  B  /\  ( y  e.  x  /\  x  C_  x ) )  ->  E. z  e.  B  ( y  e.  z  /\  z  C_  x ) )
113, 6, 10syl2anc 661 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  A  /\  y  e.  x )  ->  E. z  e.  B  ( y  e.  z  /\  z  C_  x
) )
12113expib 1191 . . . 4  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  y  e.  x
)  ->  E. z  e.  B  ( y  e.  z  /\  z  C_  x ) ) )
1312ralrimivv 2907 . . 3  |-  ( A 
C_  B  ->  A. x  e.  A  A. y  e.  x  E. z  e.  B  ( y  e.  z  /\  z  C_  x ) )
14133ad2ant2 1010 . 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
1715, 16isfne2 28686 . . 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 ) ) ) )
18173ad2ant1 1009 . 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 ) ) ) )
191, 14, 18mpbir2and 913 1  |-  ( ( B  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  A Fne B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2796   E.wrex 2797    C_ wss 3431   U.cuni 4194   class class class wbr 4395   Fnecfne 28674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-topgen 14496  df-fne 28678
This theorem is referenced by:  fnessref  28708  refssfne  28709
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