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Theorem tlt3 30322
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 2797 . . . 4 (le‘𝐾) = (le‘𝐾)
3 tlt3.l . . . 4 < = (lt‘𝐾)
41, 2, 3tlt2 30321 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌𝑌 < 𝑋))
5 tospos 30315 . . . . 5 (𝐾 ∈ Toset → 𝐾 ∈ Poset)
61, 2, 3pleval2 17408 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
7 orcom 865 . . . . . 6 ((𝑋 < 𝑌𝑋 = 𝑌) ↔ (𝑋 = 𝑌𝑋 < 𝑌))
86, 7syl6bb 288 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌𝑋 < 𝑌)))
95, 8syl3an1 1156 . . . 4 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌𝑋 < 𝑌)))
109orbi1d 911 . . 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 1081 . 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 842  w3o 1079  w3a 1080   = wceq 1525  wcel 2083   class class class wbr 4968  cfv 6232  Basecbs 16316  lecple 16405  Posetcpo 17383  ltcplt 17384  Tosetctos 17476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-iota 6196  df-fun 6234  df-fv 6240  df-proset 17371  df-poset 17389  df-plt 17401  df-toset 17477
This theorem is referenced by:  archirngz  30452  archiabllem1b  30455  archiabllem2b  30459
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