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Theorem lubel 15869
Description: An element of a set is less than or equal to the least upper bound of the set. (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
lublem.b  |-  B  =  ( Base `  K
)
lublem.l  |-  .<_  =  ( le `  K )
lublem.u  |-  U  =  ( lub `  K
)
Assertion
Ref Expression
lubel  |-  ( ( K  e.  CLat  /\  X  e.  S  /\  S  C_  B )  ->  X  .<_  ( U `  S
) )

Proof of Theorem lubel
StepHypRef Expression
1 clatl 15863 . . . 4  |-  ( K  e.  CLat  ->  K  e. 
Lat )
2 ssel 3411 . . . . 5  |-  ( S 
C_  B  ->  ( X  e.  S  ->  X  e.  B ) )
32impcom 428 . . . 4  |-  ( ( X  e.  S  /\  S  C_  B )  ->  X  e.  B )
4 lublem.b . . . . 5  |-  B  =  ( Base `  K
)
5 lublem.u . . . . 5  |-  U  =  ( lub `  K
)
64, 5lubsn 15841 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( U `  { X } )  =  X )
71, 3, 6syl2an 475 . . 3  |-  ( ( K  e.  CLat  /\  ( X  e.  S  /\  S  C_  B ) )  ->  ( U `  { X } )  =  X )
873impb 1190 . 2  |-  ( ( K  e.  CLat  /\  X  e.  S  /\  S  C_  B )  ->  ( U `  { X } )  =  X )
9 snssi 4088 . . . 4  |-  ( X  e.  S  ->  { X }  C_  S )
10 lublem.l . . . . 5  |-  .<_  =  ( le `  K )
114, 10, 5lubss 15868 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  { X }  C_  S )  -> 
( U `  { X } )  .<_  ( U `
 S ) )
129, 11syl3an3 1261 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  X  e.  S )  ->  ( U `  { X } )  .<_  ( U `
 S ) )
13123com23 1200 . 2  |-  ( ( K  e.  CLat  /\  X  e.  S  /\  S  C_  B )  ->  ( U `  { X } )  .<_  ( U `
 S ) )
148, 13eqbrtrrd 4389 1  |-  ( ( K  e.  CLat  /\  X  e.  S  /\  S  C_  B )  ->  X  .<_  ( U `  S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    C_ wss 3389   {csn 3944   class class class wbr 4367   ` cfv 5496   Basecbs 14634   lecple 14709   lubclub 15688   Latclat 15792   CLatccla 15854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-preset 15674  df-poset 15692  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-lat 15793  df-clat 15855
This theorem is referenced by:  lubun  15870  atlatmstc  35457  2polssN  36052
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