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Theorem luble 14443
Description: The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.)
Hypotheses
Ref Expression
luble.b  |-  B  =  ( Base `  K
)
luble.l  |-  .<_  =  ( le `  K )
luble.u  |-  U  =  ( lub `  K
)
Assertion
Ref Expression
luble  |-  ( ( ( K  e.  A  /\  S  C_  B )  /\  ( ( U `
 S )  e.  B  /\  X  e.  S ) )  ->  X  .<_  ( U `  S ) )

Proof of Theorem luble
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 luble.b . . . . . 6  |-  B  =  ( Base `  K
)
2 luble.l . . . . . 6  |-  .<_  =  ( le `  K )
3 luble.u . . . . . 6  |-  U  =  ( lub `  K
)
41, 2, 3lubprop 14442 . . . . 5  |-  ( ( K  e.  A  /\  S  C_  B  /\  ( U `  S )  e.  B )  ->  ( A. y  e.  S  y  .<_  ( U `  S )  /\  A. z  e.  B  ( A. y  e.  S  y  .<_  z  ->  ( U `  S )  .<_  z ) ) )
54simpld 447 . . . 4  |-  ( ( K  e.  A  /\  S  C_  B  /\  ( U `  S )  e.  B )  ->  A. y  e.  S  y  .<_  ( U `  S ) )
6 breq1 4218 . . . . 5  |-  ( y  =  X  ->  (
y  .<_  ( U `  S )  <->  X  .<_  ( U `  S ) ) )
76rspccv 3051 . . . 4  |-  ( A. y  e.  S  y  .<_  ( U `  S
)  ->  ( X  e.  S  ->  X  .<_  ( U `  S ) ) )
85, 7syl 16 . . 3  |-  ( ( K  e.  A  /\  S  C_  B  /\  ( U `  S )  e.  B )  ->  ( X  e.  S  ->  X 
.<_  ( U `  S
) ) )
983exp 1153 . 2  |-  ( K  e.  A  ->  ( S  C_  B  ->  (
( U `  S
)  e.  B  -> 
( X  e.  S  ->  X  .<_  ( U `  S ) ) ) ) )
109imp43 580 1  |-  ( ( ( K  e.  A  /\  S  C_  B )  /\  ( ( U `
 S )  e.  B  /\  X  e.  S ) )  ->  X  .<_  ( U `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   class class class wbr 4215   ` cfv 5457   Basecbs 13474   lecple 13541   lubclub 14404
This theorem is referenced by:  ple1  14478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-undef 6546  df-riota 6552  df-lub 14436
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