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Theorem posref 17553
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 17551 . 2 (𝐾 ∈ Poset → 𝐾 ∈ Proset )
2 posi.b . . 3 𝐵 = (Base‘𝐾)
3 posi.l . . 3 = (le‘𝐾)
42, 3prsref 17534 . 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 396   = wceq 1530  wcel 2107   class class class wbr 5062  cfv 6351  Basecbs 16475  lecple 16564   Proset cproset 17528  Posetcpo 17542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-nul 5206
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-iota 6311  df-fv 6359  df-proset 17530  df-poset 17548
This theorem is referenced by:  posasymb  17554  pleval2  17567  pltval3  17569  pospo  17575  lublecllem  17590  latref  17655  odupos  17737  omndmul2  30627  omndmul  30629  gsumle  30639  archirngz  30732  cvrnbtwn2  36278  cvrnbtwn3  36279  cvrnbtwn4  36282  cvrcmp  36286  llncmp  36525  lplncmp  36565  lvolcmp  36620
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