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Theorem ple1 18488
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 18417 . 2 (𝜑𝑋 (𝑈𝐵))
8 ple1.1 . . . 4 1 = (1.‘𝐾)
91, 3, 8p1val 18486 . . 3 (𝐾𝑉1 = (𝑈𝐵))
104, 9syl 17 . 2 (𝜑1 = (𝑈𝐵))
117, 10breqtrrd 5176 1 (𝜑𝑋 1 )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106   class class class wbr 5148  dom cdm 5689  cfv 6563  Basecbs 17245  lecple 17305  lubclub 18367  1.cp1 18482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-lub 18404  df-p1 18484
This theorem is referenced by:  ople1  39173  lhp2lt  39984
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