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Theorem pleval2 16788
Description: Less-than-or-equal in terms of less-than. (sspss 3668 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 16787 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 < 𝑌𝑋 = 𝑌)))
543adant1 1072 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 < 𝑌𝑋 = 𝑌)))
62, 3pltle 16784 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌𝑋 𝑌))
71, 2posref 16774 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
873adant3 1074 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑋 𝑋)
9 breq2 4587 . . . 4 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
108, 9syl5ibcom 234 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌𝑋 𝑌))
116, 10jaod 394 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 < 𝑌𝑋 = 𝑌) → 𝑋 𝑌))
125, 11impbid 201 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382  w3a 1031   = wceq 1475  wcel 1977   class class class wbr 4583  cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763  ltcplt 16764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-preset 16751  df-poset 16769  df-plt 16781
This theorem is referenced by:  pltletr  16794  plelttr  16795  tosso  16859  tlt3  28996  orngsqr  29135
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