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Theorem luble 14214
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 14213 . . . . 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 445 . . . 4  |-  ( ( K  e.  A  /\  S  C_  B  /\  ( U `  S )  e.  B )  ->  A. y  e.  S  y  .<_  ( U `  S ) )
6 breq1 4107 . . . . 5  |-  ( y  =  X  ->  (
y  .<_  ( U `  S )  <->  X  .<_  ( U `  S ) ) )
76rspccv 2957 . . . 4  |-  ( A. y  e.  S  y  .<_  ( U `  S
)  ->  ( X  e.  S  ->  X  .<_  ( U `  S ) ) )
85, 7syl 15 . . 3  |-  ( ( K  e.  A  /\  S  C_  B  /\  ( U `  S )  e.  B )  ->  ( X  e.  S  ->  X 
.<_  ( U `  S
) ) )
983exp 1150 . 2  |-  ( K  e.  A  ->  ( S  C_  B  ->  (
( U `  S
)  e.  B  -> 
( X  e.  S  ->  X  .<_  ( U `  S ) ) ) ) )
109imp43 578 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619    C_ wss 3228   class class class wbr 4104   ` cfv 5337   Basecbs 13245   lecple 13312   lubclub 14175
This theorem is referenced by:  ple1  14249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-undef 6385  df-riota 6391  df-lub 14207
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