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Theorem pltval3 17565
Description: Alternate expression for the "less than" relation. (dfpss3 4060 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
pleval2.b 𝐵 = (Base‘𝐾)
pleval2.l = (le‘𝐾)
pleval2.s < = (lt‘𝐾)
Assertion
Ref Expression
pltval3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))

Proof of Theorem pltval3
StepHypRef Expression
1 pleval2.l . . 3 = (le‘𝐾)
2 pleval2.s . . 3 < = (lt‘𝐾)
31, 2pltval 17558 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
4 pleval2.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
54, 1posref 17549 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
653adant3 1124 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑋 𝑋)
7 breq1 5060 . . . . . . 7 (𝑋 = 𝑌 → (𝑋 𝑋𝑌 𝑋))
86, 7syl5ibcom 246 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌𝑌 𝑋))
98adantr 481 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋 = 𝑌𝑌 𝑋))
104, 1posasymb 17550 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
1110biimpd 230 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌))
1211expdimp 453 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑌 𝑋𝑋 = 𝑌))
139, 12impbid 213 . . . 4 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋 = 𝑌𝑌 𝑋))
1413necon3abid 3049 . . 3 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋𝑌 ↔ ¬ 𝑌 𝑋))
1514pm5.32da 579 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑋𝑌) ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
163, 15bitrd 280 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560  Posetcpo 17538  ltcplt 17539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-proset 17526  df-poset 17544  df-plt 17556
This theorem is referenced by:  tltnle  30576  opltcon3b  36220
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