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Theorem tlt3 32960
Description: In a Toset, two elements must compare strictly, or be equal. (Contributed by Thierry Arnoux, 13-Apr-2018.)
Hypotheses
Ref Expression
tlt3.b 𝐵 = (Base‘𝐾)
tlt3.l < = (lt‘𝐾)
Assertion
Ref Expression
tlt3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌𝑋 < 𝑌𝑌 < 𝑋))

Proof of Theorem tlt3
StepHypRef Expression
1 tlt3.b . . . 4 𝐵 = (Base‘𝐾)
2 eqid 2737 . . . 4 (le‘𝐾) = (le‘𝐾)
3 tlt3.l . . . 4 < = (lt‘𝐾)
41, 2, 3tlt2 32959 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌𝑌 < 𝑋))
5 tospos 18465 . . . . 5 (𝐾 ∈ Toset → 𝐾 ∈ Poset)
61, 2, 3pleval2 18382 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
7 orcom 871 . . . . . 6 ((𝑋 < 𝑌𝑋 = 𝑌) ↔ (𝑋 = 𝑌𝑋 < 𝑌))
86, 7bitrdi 287 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌𝑋 < 𝑌)))
95, 8syl3an1 1164 . . . 4 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌𝑋 < 𝑌)))
109orbi1d 917 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(le‘𝐾)𝑌𝑌 < 𝑋) ↔ ((𝑋 = 𝑌𝑋 < 𝑌) ∨ 𝑌 < 𝑋)))
114, 10mpbid 232 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 = 𝑌𝑋 < 𝑌) ∨ 𝑌 < 𝑋))
12 df-3or 1088 . 2 ((𝑋 = 𝑌𝑋 < 𝑌𝑌 < 𝑋) ↔ ((𝑋 = 𝑌𝑋 < 𝑌) ∨ 𝑌 < 𝑋))
1311, 12sylibr 234 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌𝑋 < 𝑌𝑌 < 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 848  w3o 1086  w3a 1087   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  Basecbs 17247  lecple 17304  Posetcpo 18353  ltcplt 18354  Tosetctos 18461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-proset 18340  df-poset 18359  df-plt 18375  df-toset 18462
This theorem is referenced by:  archirngz  33196  archiabllem1b  33199  archiabllem2b  33203
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