MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ple1 Structured version   Unicode version

Theorem ple1 15543
Description: Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.)
Hypotheses
Ref Expression
ple1.b  |-  B  =  ( Base `  K
)
ple1.u  |-  U  =  ( lub `  K
)
ple1.l  |-  .<_  =  ( le `  K )
ple1.1  |-  .1.  =  ( 1. `  K )
ple1.k  |-  ( ph  ->  K  e.  V )
ple1.x  |-  ( ph  ->  X  e.  B )
ple1.d  |-  ( ph  ->  B  e.  dom  U
)
Assertion
Ref Expression
ple1  |-  ( ph  ->  X  .<_  .1.  )

Proof of Theorem ple1
StepHypRef Expression
1 ple1.b . . 3  |-  B  =  ( Base `  K
)
2 ple1.l . . 3  |-  .<_  =  ( le `  K )
3 ple1.u . . 3  |-  U  =  ( lub `  K
)
4 ple1.k . . 3  |-  ( ph  ->  K  e.  V )
5 ple1.d . . 3  |-  ( ph  ->  B  e.  dom  U
)
6 ple1.x . . 3  |-  ( ph  ->  X  e.  B )
71, 2, 3, 4, 5, 6luble 15486 . 2  |-  ( ph  ->  X  .<_  ( U `  B ) )
8 ple1.1 . . . 4  |-  .1.  =  ( 1. `  K )
91, 3, 8p1val 15541 . . 3  |-  ( K  e.  V  ->  .1.  =  ( U `  B ) )
104, 9syl 16 . 2  |-  ( ph  ->  .1.  =  ( U `
 B ) )
117, 10breqtrrd 4459 1  |-  ( ph  ->  X  .<_  .1.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802   class class class wbr 4433   dom cdm 4985   ` cfv 5574   Basecbs 14504   lecple 14576   lubclub 15440   1.cp1 15537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-lub 15473  df-p1 15539
This theorem is referenced by:  ople1  34618  lhp2lt  35427
  Copyright terms: Public domain W3C validator