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Theorem lubel 15410
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 15404 . . . 4  |-  ( K  e.  CLat  ->  K  e. 
Lat )
2 ssel 3457 . . . . 5  |-  ( S 
C_  B  ->  ( X  e.  S  ->  X  e.  B ) )
32impcom 430 . . . 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 15382 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( U `  { X } )  =  X )
71, 3, 6syl2an 477 . . 3  |-  ( ( K  e.  CLat  /\  ( X  e.  S  /\  S  C_  B ) )  ->  ( U `  { X } )  =  X )
873impb 1184 . 2  |-  ( ( K  e.  CLat  /\  X  e.  S  /\  S  C_  B )  ->  ( U `  { X } )  =  X )
9 snssi 4124 . . . 4  |-  ( X  e.  S  ->  { X }  C_  S )
10 lublem.l . . . . 5  |-  .<_  =  ( le `  K )
114, 10, 5lubss 15409 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  { X }  C_  S )  -> 
( U `  { X } )  .<_  ( U `
 S ) )
129, 11syl3an3 1254 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  X  e.  S )  ->  ( U `  { X } )  .<_  ( U `
 S ) )
13123com23 1194 . 2  |-  ( ( K  e.  CLat  /\  X  e.  S  /\  S  C_  B )  ->  ( U `  { X } )  .<_  ( U `
 S ) )
148, 13eqbrtrrd 4421 1  |-  ( ( K  e.  CLat  /\  X  e.  S  /\  S  C_  B )  ->  X  .<_  ( U `  S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3435   {csn 3984   class class class wbr 4399   ` cfv 5525   Basecbs 14291   lecple 14363   lubclub 15230   Latclat 15333   CLatccla 15395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-poset 15234  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-lat 15334  df-clat 15396
This theorem is referenced by:  lubun  15411  atlatmstc  33287  2polssN  33882
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