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Theorem pleval2 18255
Description: "Less than or equal to" in terms of "less than". (sspss 4079 analog.) (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
pleval2.b 𝐡 = (Baseβ€˜πΎ)
pleval2.l ≀ = (leβ€˜πΎ)
pleval2.s < = (ltβ€˜πΎ)
Assertion
Ref Expression
pleval2 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ (𝑋 < π‘Œ ∨ 𝑋 = π‘Œ)))
Proof of Theorem pleval2
StepHypRef Expression
1 pleval2.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 pleval2.l . . . 4 ≀ = (leβ€˜πΎ)
3 pleval2.s . . . 4 < = (ltβ€˜πΎ)
41, 2, 3pleval2i 18254 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ (𝑋 < π‘Œ ∨ 𝑋 = π‘Œ)))
543adant1 1130 . 2 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ (𝑋 < π‘Œ ∨ 𝑋 = π‘Œ)))
62, 3pltle 18251 . . 3 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 < π‘Œ β†’ 𝑋 ≀ π‘Œ))
71, 2posref 18236 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ≀ 𝑋)
873adant3 1132 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ≀ 𝑋)
9 breq2 5129 . . . 4 (𝑋 = π‘Œ β†’ (𝑋 ≀ 𝑋 ↔ 𝑋 ≀ π‘Œ))
108, 9syl5ibcom 244 . . 3 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 = π‘Œ β†’ 𝑋 ≀ π‘Œ))
116, 10jaod 857 . 2 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 < π‘Œ ∨ 𝑋 = π‘Œ) β†’ 𝑋 ≀ π‘Œ))
125, 11impbid 211 1 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ (𝑋 < π‘Œ ∨ 𝑋 = π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5125  β€˜cfv 6516  Basecbs 17109  lecple 17169  Posetcpo 18225  ltcplt 18226
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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404
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-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-iota 6468  df-fun 6518  df-fv 6524  df-proset 18213  df-poset 18231  df-plt 18248
This theorem is referenced by:  pltletr  18261  plelttr  18262  tosso  18337  tlt3  31934  orngsqr  32204
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