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Theorem pltval3 16883
Description: Alternate expression for less-than relation. (dfpss3 3676 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 16876 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
4 pleval2.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
54, 1posref 16867 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
653adant3 1079 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑋 𝑋)
7 breq1 4621 . . . . . . 7 (𝑋 = 𝑌 → (𝑋 𝑋𝑌 𝑋))
86, 7syl5ibcom 235 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌𝑌 𝑋))
98adantr 481 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋 = 𝑌𝑌 𝑋))
104, 1posasymb 16868 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
1110biimpd 219 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌))
1211expdimp 453 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑌 𝑋𝑋 = 𝑌))
139, 12impbid 202 . . . 4 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋 = 𝑌𝑌 𝑋))
1413necon3abid 2832 . . 3 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋𝑌 ↔ ¬ 𝑌 𝑋))
1514pm5.32da 672 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑋𝑌) ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
163, 15bitrd 268 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796   class class class wbr 4618  cfv 5850  Basecbs 15776  lecple 15864  Posetcpo 16856  ltcplt 16857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-preset 16844  df-poset 16862  df-plt 16874
This theorem is referenced by:  tltnle  29439  opltcon3b  33957
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