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Theorem ple1 18476
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 (𝜑𝐾𝑉)
ple1.x (𝜑𝑋𝐵)
ple1.d (𝜑𝐵 ∈ dom 𝑈)
Assertion
Ref Expression
ple1 (𝜑𝑋 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 (𝜑𝐾𝑉)
5 ple1.d . . 3 (𝜑𝐵 ∈ dom 𝑈)
6 ple1.x . . 3 (𝜑𝑋𝐵)
71, 2, 3, 4, 5, 6luble 18405 . 2 (𝜑𝑋 (𝑈𝐵))
8 ple1.1 . . . 4 1 = (1.‘𝐾)
91, 3, 8p1val 18474 . . 3 (𝐾𝑉1 = (𝑈𝐵))
104, 9syl 17 . 2 (𝜑1 = (𝑈𝐵))
117, 10breqtrrd 5170 1 (𝜑𝑋 1 )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107   class class class wbr 5142  dom cdm 5684  cfv 6560  Basecbs 17248  lecple 17305  lubclub 18356  1.cp1 18470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-lub 18392  df-p1 18472
This theorem is referenced by:  ople1  39193  lhp2lt  40004
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