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Theorem tlt3 29492
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 2621 . . . 4 (le‘𝐾) = (le‘𝐾)
3 tlt3.l . . . 4 < = (lt‘𝐾)
41, 2, 3tlt2 29491 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌𝑌 < 𝑋))
5 tospos 29485 . . . . 5 (𝐾 ∈ Toset → 𝐾 ∈ Poset)
61, 2, 3pleval2 16905 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
7 orcom 402 . . . . . 6 ((𝑋 < 𝑌𝑋 = 𝑌) ↔ (𝑋 = 𝑌𝑋 < 𝑌))
86, 7syl6bb 276 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌𝑋 < 𝑌)))
95, 8syl3an1 1356 . . . 4 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 = 𝑌𝑋 < 𝑌)))
109orbi1d 738 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(le‘𝐾)𝑌𝑌 < 𝑋) ↔ ((𝑋 = 𝑌𝑋 < 𝑌) ∨ 𝑌 < 𝑋)))
114, 10mpbid 222 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 = 𝑌𝑋 < 𝑌) ∨ 𝑌 < 𝑋))
12 df-3or 1037 . 2 ((𝑋 = 𝑌𝑋 < 𝑌𝑌 < 𝑋) ↔ ((𝑋 = 𝑌𝑋 < 𝑌) ∨ 𝑌 < 𝑋))
1311, 12sylibr 224 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌𝑋 < 𝑌𝑌 < 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  w3o 1035  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4623  cfv 5857  Basecbs 15800  lecple 15888  Posetcpo 16880  ltcplt 16881  Tosetctos 16973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-preset 16868  df-poset 16886  df-plt 16898  df-toset 16974
This theorem is referenced by:  archirngz  29570  archiabllem1b  29573  archiabllem2b  29577
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