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Theorem posref 18313
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 18311 . 2 (𝐾 ∈ Poset → 𝐾 ∈ Proset )
2 posi.b . . 3 𝐵 = (Base‘𝐾)
3 posi.l . . 3 = (le‘𝐾)
42, 3prsref 18294 . 2 ((𝐾 ∈ Proset ∧ 𝑋𝐵) → 𝑋 𝑋)
51, 4sylan 578 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098   class class class wbr 5149  cfv 6549  Basecbs 17183  lecple 17243   Proset cproset 18288  Posetcpo 18302
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-proset 18290  df-poset 18308
This theorem is referenced by:  posasymb  18314  odupos  18323  pleval2  18332  pltval3  18334  pospo  18340  lublecllem  18355  latref  18436  omndmul2  32882  omndmul  32884  gsumle  32894  archirngz  32989  cvrnbtwn2  38877  cvrnbtwn3  38878  cvrnbtwn4  38881  cvrcmp  38885  llncmp  39125  lplncmp  39165  lvolcmp  39220  lubprlem  48167  posjidm  48177  posmidm  48178
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