Theorem trleile 30655

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 2823	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	tleile 30650	. . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)𝑌 ∨ 𝑌(le‘𝐾)𝑋))
4		3simpc 1146	. . . . . 6 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
5		brxp 5603	. . . . . 6 ⊢ (𝑋(𝐵 × 𝐵)𝑌 ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
6	4, 5	sylibr 236	. . . . 5 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋(𝐵 × 𝐵)𝑌)
7		brin 5120	. . . . . 6 ⊢ (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ (𝑋(le‘𝐾)𝑌 ∧ 𝑋(𝐵 × 𝐵)𝑌))
8	7	rbaib 541	. . . . 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 464	. . . . . 6 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
11		brxp 5603	. . . . . 6 ⊢ (𝑌(𝐵 × 𝐵)𝑋 ↔ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
12	10, 11	sylibr 236	. . . . 5 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(𝐵 × 𝐵)𝑋)
13		brin 5120	. . . . . 6 ⊢ (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ (𝑌(le‘𝐾)𝑋 ∧ 𝑌(𝐵 × 𝐵)𝑋))
14	13	rbaib 541	. . . . 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 915	. . 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 259	. 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 5074	. . 3 ⊢ (𝑋 ≤ 𝑌 ↔ 𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌)
20	18	breqi 5074	. . 3 ⊢ (𝑌 ≤ 𝑋 ↔ 𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋)
21	19, 20	orbi12i 911	. 2 ⊢ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) ↔ (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ∨ 𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋))
22	17, 21	sylibr 236	1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ∩ cin 3937   class class class wbr 5068   × cxp 5555  ‘cfv 6357  Basecbs 16485  lecple 16574  Tosetctos 17645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-iota 6316  df-fv 6365  df-toset 17646

This theorem is referenced by:  ordtrest2NEWlem  31167  ordtconnlem1  31169