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Theorem posref 18275
Description: A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011.) (Proof shortened by OpenAI, 25-Mar-2020.)
Hypotheses
Ref Expression
posi.b 𝐡 = (Baseβ€˜πΎ)
posi.l ≀ = (leβ€˜πΎ)
Assertion
Ref Expression
posref ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ≀ 𝑋)

Proof of Theorem posref
StepHypRef Expression
1 posprs 18273 . 2 (𝐾 ∈ Poset β†’ 𝐾 ∈ Proset )
2 posi.b . . 3 𝐡 = (Baseβ€˜πΎ)
3 posi.l . . 3 ≀ = (leβ€˜πΎ)
42, 3prsref 18256 . 2 ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ≀ 𝑋)
51, 4sylan 580 1 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ≀ 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  β€˜cfv 6543  Basecbs 17148  lecple 17208   Proset cproset 18250  Posetcpo 18264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-proset 18252  df-poset 18270
This theorem is referenced by:  posasymb  18276  odupos  18285  pleval2  18294  pltval3  18296  pospo  18302  lublecllem  18317  latref  18398  omndmul2  32488  omndmul  32490  gsumle  32500  archirngz  32593  cvrnbtwn2  38448  cvrnbtwn3  38449  cvrnbtwn4  38452  cvrcmp  38456  llncmp  38696  lplncmp  38736  lvolcmp  38791  lubprlem  47683  posjidm  47693  posmidm  47694
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