Theorem trleile 33050

Description: In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.)

Hypotheses

Ref	Expression
trleile.b	⊢ 𝐵 = (Base‘𝐾)
trleile.l	⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))

Assertion

Ref	Expression
trleile	⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))

Proof of Theorem trleile

Step	Hyp	Ref	Expression
1		trleile.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2739	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	tleile 18376	. . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ∨ 𝑌(le‘𝐾)𝑋))
4		3simpc 1156	. . . . . 6 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
5		brxp 5667	. . . . . 6 ⊢ (𝑋(𝐵 × 𝐵)𝑌 ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
6	4, 5	sylibr 235	. . . . 5 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋(𝐵 × 𝐵)𝑌)
7		brin 5124	. . . . . 6 ⊢ (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ (𝑋(le‘𝐾)𝑌 ∧ 𝑋(𝐵 × 𝐵)𝑌))
8	7	rbaib 543	. . . . 5 ⊢ (𝑋(𝐵 × 𝐵)𝑌 → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ 𝑋(le‘𝐾)𝑌))
9	6, 8	syl 17	. . . 4 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ 𝑋(le‘𝐾)𝑌))
10	4	ancomd 462	. . . . . 6 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
11		brxp 5667	. . . . . 6 ⊢ (𝑌(𝐵 × 𝐵)𝑋 ↔ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
12	10, 11	sylibr 235	. . . . 5 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(𝐵 × 𝐵)𝑋)
13		brin 5124	. . . . . 6 ⊢ (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ (𝑌(le‘𝐾)𝑋 ∧ 𝑌(𝐵 × 𝐵)𝑋))
14	13	rbaib 543	. . . . 5 ⊢ (𝑌(𝐵 × 𝐵)𝑋 → (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ 𝑌(le‘𝐾)𝑋))
15	12, 14	syl 17	. . . 4 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ 𝑌(le‘𝐾)𝑋))
16	9, 15	orbi12d 924	. . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ∨ 𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋) ↔ (𝑋(le‘𝐾)𝑌 ∨ 𝑌(le‘𝐾)𝑋)))
17	3, 16	mpbird 258	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ∨ 𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋))
18		trleile.l	. . . 4 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
19	18	breqi 5078	. . 3 ⊢ (𝑋 ≤ 𝑌 ↔ 𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌)
20	18	breqi 5078	. . 3 ⊢ (𝑌 ≤ 𝑋 ↔ 𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋)
21	19, 20	orbi12i 920	. 2 ⊢ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) ↔ (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ∨ 𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋))
22	17, 21	sylibr 235	1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ∩ cin 3882   class class class wbr 5072   × cxp 5616  ‘cfv 6485  Basecbs 17170  lecple 17218  Tosetctos 18371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-iota 6441  df-fv 6493  df-toset 18372

This theorem is referenced by:  ordtrest2NEWlem  34106  ordtconnlem1  34108