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

Theorem prsref 17537
 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 1125 . . 3 (𝑋𝐵 → (𝑋𝐵𝑋𝐵𝑋𝐵))
3 isprs.b . . . 4 𝐵 = (Base‘𝐾)
4 isprs.l . . . 4 = (le‘𝐾)
53, 4prslem 17536 . . 3 ((𝐾 ∈ Proset ∧ (𝑋𝐵𝑋𝐵𝑋𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑋𝑋 𝑋) → 𝑋 𝑋)))
62, 5sylan2 595 . 2 ((𝐾 ∈ Proset ∧ 𝑋𝐵) → (𝑋 𝑋 ∧ ((𝑋 𝑋𝑋 𝑋) → 𝑋 𝑋)))
76simpld 498 1 ((𝐾 ∈ Proset ∧ 𝑋𝐵) → 𝑋 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5031  ‘cfv 6325  Basecbs 16478  lecple 16567   Proset cproset 17531 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-iota 6284  df-fv 6333  df-proset 17533 This theorem is referenced by:  posref  17556  mgccole1  30708  mgccole2  30709  prsdm  31282  prsrn  31283
 Copyright terms: Public domain W3C validator