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Theorem posref 18364
Description: A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011.) (Proof shortened by OpenAI, 25-Mar-2020.)
Hypotheses
Ref Expression
posi.b 𝐵 = (Base‘𝐾)
posi.l = (le‘𝐾)
Assertion
Ref Expression
posref ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)

Proof of Theorem posref
StepHypRef Expression
1 posprs 18362 . 2 (𝐾 ∈ Poset → 𝐾 ∈ Proset )
2 posi.b . . 3 𝐵 = (Base‘𝐾)
3 posi.l . . 3 = (le‘𝐾)
42, 3prsref 18344 . 2 ((𝐾 ∈ Proset ∧ 𝑋𝐵) → 𝑋 𝑋)
51, 4sylan 580 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  Basecbs 17247  lecple 17304   Proset cproset 18338  Posetcpo 18353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-proset 18340  df-poset 18359
This theorem is referenced by:  posasymb  18365  odupos  18373  pleval2  18382  pltval3  18384  pospo  18390  lublecllem  18405  latref  18486  omndmul2  33089  omndmul  33091  gsumle  33101  archirngz  33196  cvrnbtwn2  39276  cvrnbtwn3  39277  cvrnbtwn4  39280  cvrcmp  39284  llncmp  39524  lplncmp  39564  lvolcmp  39619  lubprlem  48859  posjidm  48869  posmidm  48870
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