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Theorem fness 30407
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 996 . 2  |-  ( ( B  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  X  =  Y )
2 ssel2 3484 . . . . . . 7  |-  ( ( A  C_  B  /\  x  e.  A )  ->  x  e.  B )
323adant3 1014 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  A  /\  y  e.  x )  ->  x  e.  B )
4 simp3 996 . . . . . . 7  |-  ( ( A  C_  B  /\  x  e.  A  /\  y  e.  x )  ->  y  e.  x )
5 ssid 3508 . . . . . . 7  |-  x  C_  x
64, 5jctir 536 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  A  /\  y  e.  x )  ->  ( y  e.  x  /\  x  C_  x ) )
7 elequ2 1828 . . . . . . . 8  |-  ( z  =  x  ->  (
y  e.  z  <->  y  e.  x ) )
8 sseq1 3510 . . . . . . . 8  |-  ( z  =  x  ->  (
z  C_  x  <->  x  C_  x
) )
97, 8anbi12d 708 . . . . . . 7  |-  ( z  =  x  ->  (
( y  e.  z  /\  z  C_  x
)  <->  ( y  e.  x  /\  x  C_  x ) ) )
109rspcev 3207 . . . . . 6  |-  ( ( x  e.  B  /\  ( y  e.  x  /\  x  C_  x ) )  ->  E. z  e.  B  ( y  e.  z  /\  z  C_  x ) )
113, 6, 10syl2anc 659 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  A  /\  y  e.  x )  ->  E. z  e.  B  ( y  e.  z  /\  z  C_  x
) )
12113expib 1197 . . . 4  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  y  e.  x
)  ->  E. z  e.  B  ( y  e.  z  /\  z  C_  x ) ) )
1312ralrimivv 2874 . . 3  |-  ( A 
C_  B  ->  A. x  e.  A  A. y  e.  x  E. z  e.  B  ( y  e.  z  /\  z  C_  x ) )
14133ad2ant2 1016 . 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 30400 . . 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 1015 . 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 920 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805    C_ wss 3461   U.cuni 4235   class class class wbr 4439   Fnecfne 30394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-topgen 14933  df-fne 30395
This theorem is referenced by:  fnessref  30415  refssfne  30416
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