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Theorem lubub 15876
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 . . . 4  |-  B  =  ( Base `  K
)
2 lublem.l . . . 4  |-  .<_  =  ( le `  K )
3 lublem.u . . . 4  |-  U  =  ( lub `  K
)
41, 2, 3lublem 15875 . . 3  |-  ( ( 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 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  A. y  e.  S  y  .<_  ( U `  S ) )
6 breq1 4459 . . 3  |-  ( y  =  X  ->  (
y  .<_  ( U `  S )  <->  X  .<_  ( U `  S ) ) )
76rspccva 3209 . 2  |-  ( ( A. y  e.  S  y  .<_  ( U `  S )  /\  X  e.  S )  ->  X  .<_  ( U `  S
) )
85, 7stoic3 1610 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 973    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   class class class wbr 4456   ` cfv 5594   Basecbs 14644   lecple 14719   lubclub 15698   CLatccla 15864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-lub 15731  df-clat 15865
This theorem is referenced by:  lubss  15878
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