Theorem trleile 33231

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 2769	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	tleile 18474	. . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ∨ 𝑌(le‘𝐾)𝑋))
4		3simpc 1166	. . . . . 6 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
5		brxp 5711	. . . . . 6 ⊢ (𝑋(𝐵 × 𝐵)𝑌 ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
6	4, 5	sylibr 237	. . . . 5 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋(𝐵 × 𝐵)𝑌)
7		brin 5167	. . . . . 6 ⊢ (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ (𝑋(le‘𝐾)𝑌 ∧ 𝑋(𝐵 × 𝐵)𝑌))
8	7	rbaib 547	. . . . 5 ⊢ (𝑋(𝐵 × 𝐵)𝑌 → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ 𝑋(le‘𝐾)𝑌))
9	6, 8	syl 18	. . . 4 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ 𝑋(le‘𝐾)𝑌))
10	4	ancomd 466	. . . . . 6 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
11		brxp 5711	. . . . . 6 ⊢ (𝑌(𝐵 × 𝐵)𝑋 ↔ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
12	10, 11	sylibr 237	. . . . 5 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(𝐵 × 𝐵)𝑋)
13		brin 5167	. . . . . 6 ⊢ (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ (𝑌(le‘𝐾)𝑋 ∧ 𝑌(𝐵 × 𝐵)𝑋))
14	13	rbaib 547	. . . . 5 ⊢ (𝑌(𝐵 × 𝐵)𝑋 → (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ 𝑌(le‘𝐾)𝑋))
15	12, 14	syl 18	. . . 4 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ 𝑌(le‘𝐾)𝑋))
16	9, 15	orbi12d 931	. . 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 260	. 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 5119	. . 3 ⊢ (𝑋 ≤ 𝑌 ↔ 𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌)
20	18	breqi 5119	. . 3 ⊢ (𝑌 ≤ 𝑋 ↔ 𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋)
21	19, 20	orbi12i 927	. 2 ⊢ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) ↔ (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ∨ 𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋))
22	17, 21	sylibr 237	1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ∩ cin 3912   class class class wbr 5113   × cxp 5660  ‘cfv 6537  Basecbs 17268  lecple 17316  Tosetctos 18469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-iota 6493  df-fv 6545  df-toset 18470

This theorem is referenced by:  ordtrest2NEWlem  34256  ordtconnlem1  34258