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Theorem prsref 18256
Description: "Less than or equal to" is reflexive in a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
isprs.b 𝐡 = (Baseβ€˜πΎ)
isprs.l ≀ = (leβ€˜πΎ)
Assertion
Ref Expression
prsref ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ≀ 𝑋)

Proof of Theorem prsref
StepHypRef Expression
1 id 22 . . . 4 (𝑋 ∈ 𝐡 β†’ 𝑋 ∈ 𝐡)
21, 1, 13jca 1128 . . 3 (𝑋 ∈ 𝐡 β†’ (𝑋 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡))
3 isprs.b . . . 4 𝐡 = (Baseβ€˜πΎ)
4 isprs.l . . . 4 ≀ = (leβ€˜πΎ)
53, 4prslem 18255 . . 3 ((𝐾 ∈ Proset ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)) β†’ (𝑋 ≀ 𝑋 ∧ ((𝑋 ≀ 𝑋 ∧ 𝑋 ≀ 𝑋) β†’ 𝑋 ≀ 𝑋)))
62, 5sylan2 593 . 2 ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ≀ 𝑋 ∧ ((𝑋 ≀ 𝑋 ∧ 𝑋 ≀ 𝑋) β†’ 𝑋 ≀ 𝑋)))
76simpld 495 1 ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ≀ 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  β€˜cfv 6543  Basecbs 17148  lecple 17208   Proset cproset 18250
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 5306
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 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-proset 18252
This theorem is referenced by:  posref  18275  mgccole1  32415  mgccole2  32416  prsdm  33180  prsrn  33181  prsthinc  47762
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