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Theorem luble 14401
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 14400 . . . . 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 446 . . . 4  |-  ( ( K  e.  A  /\  S  C_  B  /\  ( U `  S )  e.  B )  ->  A. y  e.  S  y  .<_  ( U `  S ) )
6 breq1 4183 . . . . 5  |-  ( y  =  X  ->  (
y  .<_  ( U `  S )  <->  X  .<_  ( U `  S ) ) )
76rspccv 3017 . . . 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 1152 . 2  |-  ( K  e.  A  ->  ( S  C_  B  ->  (
( U `  S
)  e.  B  -> 
( X  e.  S  ->  X  .<_  ( U `  S ) ) ) ) )
109imp43 579 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674    C_ wss 3288   class class class wbr 4180   ` cfv 5421   Basecbs 13432   lecple 13499   lubclub 14362
This theorem is referenced by:  ple1  14436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-undef 6510  df-riota 6516  df-lub 14394
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