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Theorem prstr 17555
 Description: "Less than or equal to" is transitive 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
prstr ((𝐾 ∈ Proset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑋 𝑍)

Proof of Theorem prstr
StepHypRef Expression
1 isprs.b . . . 4 𝐵 = (Base‘𝐾)
2 isprs.l . . . 4 = (le‘𝐾)
31, 2prslem 17553 . . 3 ((𝐾 ∈ Proset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
43simprd 499 . 2 ((𝐾 ∈ Proset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))
543impia 1114 1 ((𝐾 ∈ Proset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑋 𝑍)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5034  ‘cfv 6332  Basecbs 16495  lecple 16584   Proset cproset 17548 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 5178 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 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-iota 6291  df-fv 6340  df-proset 17550 This theorem is referenced by:  drsdirfi  17560  mgcmnt1  30744  mgcmnt2  30745  mgcmntco  30746  dfmgc2lem  30747
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