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Theorem tleile 30575
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 1129 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
2 simp3 1130 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
3 tleile.b . . . . 5 𝐵 = (Base‘𝐾)
4 tleile.l . . . . 5 = (le‘𝐾)
53, 4istos 17633 . . . 4 (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
65simprbi 497 . . 3 (𝐾 ∈ Toset → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))
763ad2ant1 1125 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))
8 breq1 5060 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
9 breq2 5061 . . . 4 (𝑥 = 𝑋 → (𝑦 𝑥𝑦 𝑋))
108, 9orbi12d 912 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑋 𝑦𝑦 𝑋)))
11 breq2 5061 . . . 4 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
12 breq1 5060 . . . 4 (𝑦 = 𝑌 → (𝑦 𝑋𝑌 𝑋))
1311, 12orbi12d 912 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 𝑋) ↔ (𝑋 𝑌𝑌 𝑋)))
1410, 13rspc2va 3631 . 2 (((𝑋𝐵𝑌𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) → (𝑋 𝑌𝑌 𝑋))
151, 2, 7, 14syl21anc 833 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 841  w3a 1079   = wceq 1528  wcel 2105  wral 3135   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560  Posetcpo 17538  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-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  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-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-iota 6307  df-fv 6356  df-toset 17632
This theorem is referenced by:  tltnle  30576  odutos  30577  trleile  30580
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