Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pleval2 Structured version   Visualization version   GIF version

Theorem pleval2 17573
 Description: "Less than or equal to" in terms of "less than". (sspss 4062 analog.) (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
pleval2.b 𝐵 = (Base‘𝐾)
pleval2.l = (le‘𝐾)
pleval2.s < = (lt‘𝐾)
Assertion
Ref Expression
pleval2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))

Proof of Theorem pleval2
StepHypRef Expression
1 pleval2.b . . . 4 𝐵 = (Base‘𝐾)
2 pleval2.l . . . 4 = (le‘𝐾)
3 pleval2.s . . . 4 < = (lt‘𝐾)
41, 2, 3pleval2i 17572 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 < 𝑌𝑋 = 𝑌)))
543adant1 1127 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 < 𝑌𝑋 = 𝑌)))
62, 3pltle 17569 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌𝑋 𝑌))
71, 2posref 17559 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
873adant3 1129 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑋 𝑋)
9 breq2 5057 . . . 4 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
108, 9syl5ibcom 248 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌𝑋 𝑌))
116, 10jaod 856 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 < 𝑌𝑋 = 𝑌) → 𝑋 𝑌))
125, 11impbid 215 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   class class class wbr 5053  ‘cfv 6344  Basecbs 16481  lecple 16570  Posetcpo 17548  ltcplt 17549 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-proset 17536  df-poset 17554  df-plt 17566 This theorem is referenced by:  pltletr  17579  plelttr  17580  tosso  17644  tlt3  30658  orngsqr  30904
 Copyright terms: Public domain W3C validator