MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  posref Structured version   Visualization version   GIF version

Theorem posref 18271
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 18269 . 2 (𝐾 ∈ Poset β†’ 𝐾 ∈ Proset )
2 posi.b . . 3 𝐡 = (Baseβ€˜πΎ)
3 posi.l . . 3 ≀ = (leβ€˜πΎ)
42, 3prsref 18252 . 2 ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ≀ 𝑋)
51, 4sylan 581 1 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ≀ 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  β€˜cfv 6544  Basecbs 17144  lecple 17204   Proset cproset 18246  Posetcpo 18260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-proset 18248  df-poset 18266
This theorem is referenced by:  posasymb  18272  odupos  18281  pleval2  18290  pltval3  18292  pospo  18298  lublecllem  18313  latref  18394  omndmul2  32230  omndmul  32232  gsumle  32242  archirngz  32335  cvrnbtwn2  38145  cvrnbtwn3  38146  cvrnbtwn4  38149  cvrcmp  38153  llncmp  38393  lplncmp  38433  lvolcmp  38488  lubprlem  47595  posjidm  47605  posmidm  47606
  Copyright terms: Public domain W3C validator