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Theorem lubel 14550
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 14544 . . . 4  |-  ( K  e.  CLat  ->  K  e. 
Lat )
2 ssel 3343 . . . . 5  |-  ( S 
C_  B  ->  ( X  e.  S  ->  X  e.  B ) )
32impcom 421 . . . 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 14524 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( U `  { X } )  =  X )
71, 3, 6syl2an 465 . . 3  |-  ( ( K  e.  CLat  /\  ( X  e.  S  /\  S  C_  B ) )  ->  ( U `  { X } )  =  X )
873impb 1150 . 2  |-  ( ( K  e.  CLat  /\  X  e.  S  /\  S  C_  B )  ->  ( U `  { X } )  =  X )
9 snssi 3943 . . . 4  |-  ( X  e.  S  ->  { X }  C_  S )
10 lublem.l . . . . 5  |-  .<_  =  ( le `  K )
114, 10, 5lubss 14549 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  { X }  C_  S )  -> 
( U `  { X } )  .<_  ( U `
 S ) )
129, 11syl3an3 1220 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  X  e.  S )  ->  ( U `  { X } )  .<_  ( U `
 S ) )
13123com23 1160 . 2  |-  ( ( K  e.  CLat  /\  X  e.  S  /\  S  C_  B )  ->  ( U `  { X } )  .<_  ( U `
 S ) )
148, 13eqbrtrrd 4235 1  |-  ( ( K  e.  CLat  /\  X  e.  S  /\  S  C_  B )  ->  X  .<_  ( U `  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3321   {csn 3815   class class class wbr 4213   ` cfv 5455   Basecbs 13470   lecple 13537   lubclub 14400   Latclat 14475   CLatccla 14537
This theorem is referenced by:  lubun  14551  lubunNEW  29772  atlatmstc  30118  2polssN  30713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-lub 14432  df-join 14434  df-meet 14435  df-lat 14476  df-clat 14538
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