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Theorem lubub 14466
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 14465 . . . 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 446 . . 3  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  A. y  e.  S  y  .<_  ( U `  S ) )
6 breq1 4149 . . . 4  |-  ( y  =  X  ->  (
y  .<_  ( U `  S )  <->  X  .<_  ( U `  S ) ) )
76rspccva 2987 . . 3  |-  ( ( A. y  e.  S  y  .<_  ( U `  S )  /\  X  e.  S )  ->  X  .<_  ( U `  S
) )
85, 7sylan 458 . 2  |-  ( ( ( K  e.  CLat  /\  S  C_  B )  /\  X  e.  S
)  ->  X  .<_  ( U `  S ) )
983impa 1148 1  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  X  e.  S )  ->  X  .<_  ( U `  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642    C_ wss 3256   class class class wbr 4146   ` cfv 5387   Basecbs 13389   lecple 13456   lubclub 14319   CLatccla 14456
This theorem is referenced by:  lubss  14468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-undef 6472  df-riota 6478  df-lub 14351  df-clat 14457
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