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Theorem tleile 30227
 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 1128 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
2 simp3 1129 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
3 tleile.b . . . . 5 𝐵 = (Base‘𝐾)
4 tleile.l . . . . 5 = (le‘𝐾)
53, 4istos 17425 . . . 4 (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
65simprbi 492 . . 3 (𝐾 ∈ Toset → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))
763ad2ant1 1124 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))
8 breq1 4891 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
9 breq2 4892 . . . 4 (𝑥 = 𝑋 → (𝑦 𝑥𝑦 𝑋))
108, 9orbi12d 905 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑋 𝑦𝑦 𝑋)))
11 breq2 4892 . . . 4 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
12 breq1 4891 . . . 4 (𝑦 = 𝑌 → (𝑦 𝑋𝑌 𝑋))
1311, 12orbi12d 905 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 𝑋) ↔ (𝑋 𝑌𝑌 𝑋)))
1410, 13rspc2va 3525 . 2 (((𝑋𝐵𝑌𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) → (𝑋 𝑌𝑌 𝑋))
151, 2, 7, 14syl21anc 828 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 836   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090   class class class wbr 4888  ‘cfv 6137  Basecbs 16259  lecple 16349  Posetcpo 17330  Tosetctos 17423 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5027 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-iota 6101  df-fv 6145  df-toset 17424 This theorem is referenced by:  tltnle  30228  odutos  30229  trleile  30232
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