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Theorem tleile 18416
Description: In a Toset, any two elements are comparable. (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 1134 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
2 simp3 1135 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
3 tleile.b . . . . 5 𝐵 = (Base‘𝐾)
4 tleile.l . . . . 5 = (le‘𝐾)
53, 4istos 18413 . . . 4 (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
65simprbi 495 . . 3 (𝐾 ∈ Toset → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))
763ad2ant1 1130 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))
8 breq1 5152 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
9 breq2 5153 . . . 4 (𝑥 = 𝑋 → (𝑦 𝑥𝑦 𝑋))
108, 9orbi12d 916 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑋 𝑦𝑦 𝑋)))
11 breq2 5153 . . . 4 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
12 breq1 5152 . . . 4 (𝑦 = 𝑌 → (𝑦 𝑋𝑌 𝑋))
1311, 12orbi12d 916 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 𝑋) ↔ (𝑋 𝑌𝑌 𝑋)))
1410, 13rspc2va 3618 . 2 (((𝑋𝐵𝑌𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) → (𝑋 𝑌𝑌 𝑋))
151, 2, 7, 14syl21anc 836 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 845  w3a 1084   = wceq 1533  wcel 2098  wral 3050   class class class wbr 5149  cfv 6549  Basecbs 17183  lecple 17243  Posetcpo 18302  Tosetctos 18411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-toset 18412
This theorem is referenced by:  tltnle  18417  odutos  32784  trleile  32787  toslat  48176
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