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Theorem fness 26385
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 957 . 2  |-  ( ( B  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  X  =  Y )
2 ssel2 3188 . . . . . . 7  |-  ( ( A  C_  B  /\  x  e.  A )  ->  x  e.  B )
323adant3 975 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  A  /\  y  e.  x )  ->  x  e.  B )
4 simp3 957 . . . . . . 7  |-  ( ( A  C_  B  /\  x  e.  A  /\  y  e.  x )  ->  y  e.  x )
5 ssid 3210 . . . . . . 7  |-  x  C_  x
64, 5jctir 524 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  A  /\  y  e.  x )  ->  ( y  e.  x  /\  x  C_  x ) )
7 elequ2 1701 . . . . . . . 8  |-  ( z  =  x  ->  (
y  e.  z  <->  y  e.  x ) )
8 sseq1 3212 . . . . . . . 8  |-  ( z  =  x  ->  (
z  C_  x  <->  x  C_  x
) )
97, 8anbi12d 691 . . . . . . 7  |-  ( z  =  x  ->  (
( y  e.  z  /\  z  C_  x
)  <->  ( y  e.  x  /\  x  C_  x ) ) )
109rspcev 2897 . . . . . 6  |-  ( ( x  e.  B  /\  ( y  e.  x  /\  x  C_  x ) )  ->  E. z  e.  B  ( y  e.  z  /\  z  C_  x ) )
113, 6, 10syl2anc 642 . . . . 5  |-  ( ( A  C_  B  /\  x  e.  A  /\  y  e.  x )  ->  E. z  e.  B  ( y  e.  z  /\  z  C_  x
) )
12113expib 1154 . . . 4  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  y  e.  x
)  ->  E. z  e.  B  ( y  e.  z  /\  z  C_  x ) ) )
1312ralrimivv 2647 . . 3  |-  ( A 
C_  B  ->  A. x  e.  A  A. y  e.  x  E. z  e.  B  ( y  e.  z  /\  z  C_  x ) )
14133ad2ant2 977 . 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 26374 . . 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 976 . 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 888 1  |-  ( ( B  e.  C  /\  A  C_  B  /\  X  =  Y )  ->  A Fne B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   U.cuni 3843   class class class wbr 4039   Fnecfne 26362
This theorem is referenced by:  fnessref  26396  refssfne  26397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-topgen 13360  df-fne 26366
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