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Theorem prsref 16755
 Description: Less-or-equal is reflexive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
isprs.b 𝐵 = (Base‘𝐾)
isprs.l = (le‘𝐾)
Assertion
Ref Expression
prsref ((𝐾 ∈ Preset ∧ 𝑋𝐵) → 𝑋 𝑋)

Proof of Theorem prsref
StepHypRef Expression
1 id 22 . . . 4 (𝑋𝐵𝑋𝐵)
21, 1, 13jca 1235 . . 3 (𝑋𝐵 → (𝑋𝐵𝑋𝐵𝑋𝐵))
3 isprs.b . . . 4 𝐵 = (Base‘𝐾)
4 isprs.l . . . 4 = (le‘𝐾)
53, 4prslem 16754 . . 3 ((𝐾 ∈ Preset ∧ (𝑋𝐵𝑋𝐵𝑋𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑋𝑋 𝑋) → 𝑋 𝑋)))
62, 5sylan2 490 . 2 ((𝐾 ∈ Preset ∧ 𝑋𝐵) → (𝑋 𝑋 ∧ ((𝑋 𝑋𝑋 𝑋) → 𝑋 𝑋)))
76simpld 474 1 ((𝐾 ∈ Preset ∧ 𝑋𝐵) → 𝑋 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775   Preset cpreset 16749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-preset 16751 This theorem is referenced by:  posref  16774  prsdm  29288  prsrn  29289
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