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Theorem prsref 18345
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 18344 . . 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 494 1 ((𝐾 ∈ Proset ∧ 𝑋𝐵) → 𝑋 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107   class class class wbr 5142  cfv 6560  Basecbs 17248  lecple 17305   Proset cproset 18339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-proset 18341
This theorem is referenced by:  posref  18365  mgccole1  32981  mgccole2  32982  prsdm  33914  prsrn  33915  prsthinc  49136
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