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Theorem tltnle 29987
Description: In a Toset, less-than is equivalent to not inverse less-than-or-equal see pltnle 17171. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Hypotheses
Ref Expression
tleile.b 𝐵 = (Base‘𝐾)
tleile.l = (le‘𝐾)
tltnle.s < = (lt‘𝐾)
Assertion
Ref Expression
tltnle ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 𝑋))

Proof of Theorem tltnle
StepHypRef Expression
1 tospos 29983 . . 3 (𝐾 ∈ Toset → 𝐾 ∈ Poset)
2 tleile.b . . . 4 𝐵 = (Base‘𝐾)
3 tleile.l . . . 4 = (le‘𝐾)
4 tltnle.s . . . 4 < = (lt‘𝐾)
52, 3, 4pltval3 17172 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
61, 5syl3an1 1195 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
72, 3tleile 29986 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
8 ibar 520 . . . 4 ((𝑋 𝑌𝑌 𝑋) → (¬ 𝑌 𝑋 ↔ ((𝑋 𝑌𝑌 𝑋) ∧ ¬ 𝑌 𝑋)))
9 pm5.61 1014 . . . 4 (((𝑋 𝑌𝑌 𝑋) ∧ ¬ 𝑌 𝑋) ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋))
108, 9syl6rbb 279 . . 3 ((𝑋 𝑌𝑌 𝑋) → ((𝑋 𝑌 ∧ ¬ 𝑌 𝑋) ↔ ¬ 𝑌 𝑋))
117, 10syl 17 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌 ∧ ¬ 𝑌 𝑋) ↔ ¬ 𝑌 𝑋))
126, 11bitrd 270 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2156   class class class wbr 4844  cfv 6101  Basecbs 16068  lecple 16160  Posetcpo 17145  ltcplt 17146  Tosetctos 17238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6064  df-fun 6103  df-fv 6109  df-proset 17133  df-poset 17151  df-plt 17163  df-toset 17239
This theorem is referenced by:  tlt2  29989  toslublem  29992  tosglblem  29994  isarchi2  30064  archirng  30067  archiabllem2c  30074  archiabl  30077
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