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Theorem lubub 15294
Description: The LUB of a complete lattice subset is an upper bound. (Contributed by NM, 19-Oct-2011.)
Hypotheses
Ref Expression
lublem.b  |-  B  =  ( Base `  K
)
lublem.l  |-  .<_  =  ( le `  K )
lublem.u  |-  U  =  ( lub `  K
)
Assertion
Ref Expression
lubub  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  X  e.  S )  ->  X  .<_  ( U `  S
) )

Proof of Theorem lubub
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lublem.b . . . . 5  |-  B  =  ( Base `  K
)
2 lublem.l . . . . 5  |-  .<_  =  ( le `  K )
3 lublem.u . . . . 5  |-  U  =  ( lub `  K
)
41, 2, 3lublem 15293 . . . 4  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( A. y  e.  S  y  .<_  ( U `  S )  /\  A. z  e.  B  ( A. y  e.  S  y  .<_  z  ->  ( U `  S )  .<_  z ) ) )
54simpld 459 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  A. y  e.  S  y  .<_  ( U `  S ) )
6 breq1 4300 . . . 4  |-  ( y  =  X  ->  (
y  .<_  ( U `  S )  <->  X  .<_  ( U `  S ) ) )
76rspccva 3077 . . 3  |-  ( ( A. y  e.  S  y  .<_  ( U `  S )  /\  X  e.  S )  ->  X  .<_  ( U `  S
) )
85, 7sylan 471 . 2  |-  ( ( ( K  e.  CLat  /\  S  C_  B )  /\  X  e.  S
)  ->  X  .<_  ( U `  S ) )
983impa 1182 1  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  X  e.  S )  ->  X  .<_  ( U `  S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720    C_ wss 3333   class class class wbr 4297   ` cfv 5423   Basecbs 14179   lecple 14250   lubclub 15117   CLatccla 15282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-lub 15149  df-clat 15283
This theorem is referenced by:  lubss  15296
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