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Theorem tlt3 30579
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 2818 . . . 4 (le‘𝐾) = (le‘𝐾)
3 tlt3.l . . . 4 < = (lt‘𝐾)
41, 2, 3tlt2 30578 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌𝑌 < 𝑋))
5 tospos 30572 . . . . 5 (𝐾 ∈ Toset → 𝐾 ∈ Poset)
61, 2, 3pleval2 17563 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
7 orcom 864 . . . . . 6 ((𝑋 < 𝑌𝑋 = 𝑌) ↔ (𝑋 = 𝑌𝑋 < 𝑌))
86, 7syl6bb 288 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌𝑋 < 𝑌)))
95, 8syl3an1 1155 . . . 4 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌𝑋 < 𝑌)))
109orbi1d 910 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(le‘𝐾)𝑌𝑌 < 𝑋) ↔ ((𝑋 = 𝑌𝑋 < 𝑌) ∨ 𝑌 < 𝑋)))
114, 10mpbid 233 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 = 𝑌𝑋 < 𝑌) ∨ 𝑌 < 𝑋))
12 df-3or 1080 . 2 ((𝑋 = 𝑌𝑋 < 𝑌𝑌 < 𝑋) ↔ ((𝑋 = 𝑌𝑋 < 𝑌) ∨ 𝑌 < 𝑋))
1311, 12sylibr 235 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌𝑋 < 𝑌𝑌 < 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wo 841  w3o 1078  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560  Posetcpo 17538  ltcplt 17539  Tosetctos 17631
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-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  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  df-toset 17632
This theorem is referenced by:  archirngz  30745  archiabllem1b  30748  archiabllem2b  30752
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