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Theorem tleile 29646
Description: In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Hypotheses
Ref Expression
tleile.b 𝐵 = (Base‘𝐾)
tleile.l = (le‘𝐾)
Assertion
Ref Expression
tleile ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))

Proof of Theorem tleile
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1061 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
2 simp3 1062 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
3 tleile.b . . . . 5 𝐵 = (Base‘𝐾)
4 tleile.l . . . . 5 = (le‘𝐾)
53, 4istos 17029 . . . 4 (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
65simprbi 480 . . 3 (𝐾 ∈ Toset → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))
763ad2ant1 1081 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))
8 breq1 4654 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
9 breq2 4655 . . . 4 (𝑥 = 𝑋 → (𝑦 𝑥𝑦 𝑋))
108, 9orbi12d 746 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑋 𝑦𝑦 𝑋)))
11 breq2 4655 . . . 4 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
12 breq1 4654 . . . 4 (𝑦 = 𝑌 → (𝑦 𝑋𝑌 𝑋))
1311, 12orbi12d 746 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 𝑋) ↔ (𝑋 𝑌𝑌 𝑋)))
1410, 13rspc2va 3321 . 2 (((𝑋𝐵𝑌𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) → (𝑋 𝑌𝑌 𝑋))
151, 2, 7, 14syl21anc 1324 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  w3a 1037   = wceq 1482  wcel 1989  wral 2911   class class class wbr 4651  cfv 5886  Basecbs 15851  lecple 15942  Posetcpo 16934  Tosetctos 17027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-toset 17028
This theorem is referenced by:  tltnle  29647  odutos  29648  trleile  29651
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