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Theorem prsref 16848
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 1240 . . 3 (𝑋𝐵 → (𝑋𝐵𝑋𝐵𝑋𝐵))
3 isprs.b . . . 4 𝐵 = (Base‘𝐾)
4 isprs.l . . . 4 = (le‘𝐾)
53, 4prslem 16847 . . 3 ((𝐾 ∈ Preset ∧ (𝑋𝐵𝑋𝐵𝑋𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑋𝑋 𝑋) → 𝑋 𝑋)))
62, 5sylan2 491 . 2 ((𝐾 ∈ Preset ∧ 𝑋𝐵) → (𝑋 𝑋 ∧ ((𝑋 𝑋𝑋 𝑋) → 𝑋 𝑋)))
76simpld 475 1 ((𝐾 ∈ Preset ∧ 𝑋𝐵) → 𝑋 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992   class class class wbr 4618  cfv 5850  Basecbs 15776  lecple 15864   Preset cpreset 16842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5813  df-fv 5858  df-preset 16844
This theorem is referenced by:  posref  16867  prsdm  29734  prsrn  29735
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