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Theorem posref 16998
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 16996 . 2 (𝐾 ∈ Poset → 𝐾 ∈ Preset )
2 posi.b . . 3 𝐵 = (Base‘𝐾)
3 posi.l . . 3 = (le‘𝐾)
42, 3prsref 16979 . 2 ((𝐾 ∈ Preset ∧ 𝑋𝐵) → 𝑋 𝑋)
51, 4sylan 487 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030   class class class wbr 4685  cfv 5926  Basecbs 15904  lecple 15995   Preset cpreset 16973  Posetcpo 16987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-preset 16975  df-poset 16993
This theorem is referenced by:  posasymb  16999  pleval2  17012  pltval3  17014  pospo  17020  lublecllem  17035  latref  17100  odupos  17182  omndmul2  29840  omndmul  29842  archirngz  29871  gsumle  29907  cvrnbtwn2  34880  cvrnbtwn3  34881  cvrnbtwn4  34884  cvrcmp  34888  llncmp  35126  lplncmp  35166  lvolcmp  35221
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