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Theorem tltnle 30635
Description: In a Toset, less-than is equivalent to not inverse "less than or equal to" see pltnle 17554. (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 30631 . . 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 17555 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
61, 5syl3an1 1160 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
72, 3tleile 30634 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
8 ibar 532 . . . 4 ((𝑋 𝑌𝑌 𝑋) → (¬ 𝑌 𝑋 ↔ ((𝑋 𝑌𝑌 𝑋) ∧ ¬ 𝑌 𝑋)))
9 pm5.61 998 . . . 4 (((𝑋 𝑌𝑌 𝑋) ∧ ¬ 𝑌 𝑋) ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋))
108, 9syl6rbb 291 . . 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 282 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115   class class class wbr 5039  cfv 6328  Basecbs 16461  lecple 16550  Posetcpo 17528  ltcplt 17529  Tosetctos 17621
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-proset 17516  df-poset 17534  df-plt 17546  df-toset 17622
This theorem is referenced by:  tlt2  30637  toslublem  30640  tosglblem  30642  isarchi2  30821  archirng  30824  archiabllem2c  30831  archiabl  30834
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