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Theorem prsref 18248
Description: "Less than or equal to" is reflexive in a proset. (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 ∈ Proset ∧ 𝑋𝐵) → 𝑋 𝑋)

Proof of Theorem prsref
StepHypRef Expression
1 id 22 . . . 4 (𝑋𝐵𝑋𝐵)
21, 1, 13jca 1128 . . 3 (𝑋𝐵 → (𝑋𝐵𝑋𝐵𝑋𝐵))
3 isprs.b . . . 4 𝐵 = (Base‘𝐾)
4 isprs.l . . . 4 = (le‘𝐾)
53, 4prslem 18247 . . 3 ((𝐾 ∈ Proset ∧ (𝑋𝐵𝑋𝐵𝑋𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑋𝑋 𝑋) → 𝑋 𝑋)))
62, 5sylan2 593 . 2 ((𝐾 ∈ Proset ∧ 𝑋𝐵) → (𝑋 𝑋 ∧ ((𝑋 𝑋𝑋 𝑋) → 𝑋 𝑋)))
76simpld 495 1 ((𝐾 ∈ Proset ∧ 𝑋𝐵) → 𝑋 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106   class class class wbr 5147  cfv 6540  Basecbs 17140  lecple 17200   Proset cproset 18242
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 5305
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 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-proset 18244
This theorem is referenced by:  posref  18267  mgccole1  32147  mgccole2  32148  prsdm  32882  prsrn  32883  prsthinc  47627
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