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Theorem posref 18376
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 18374 . 2 (𝐾 ∈ Poset → 𝐾 ∈ Proset )
2 posi.b . . 3 𝐵 = (Base‘𝐾)
3 posi.l . . 3 = (le‘𝐾)
42, 3prsref 18356 . 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 1537  wcel 2106   class class class wbr 5148  cfv 6563  Basecbs 17245  lecple 17305   Proset cproset 18350  Posetcpo 18365
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-ext 2706  ax-nul 5312
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-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-proset 18352  df-poset 18371
This theorem is referenced by:  posasymb  18377  odupos  18386  pleval2  18395  pltval3  18397  pospo  18403  lublecllem  18418  latref  18499  omndmul2  33072  omndmul  33074  gsumle  33084  archirngz  33179  cvrnbtwn2  39257  cvrnbtwn3  39258  cvrnbtwn4  39261  cvrcmp  39265  llncmp  39505  lplncmp  39545  lvolcmp  39600  lubprlem  48759  posjidm  48769  posmidm  48770
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