Theorem le2tri3i 11343

Description: Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)

Hypotheses

Ref	Expression
lt.1	⊢ 𝐴 ∈ ℝ
lt.2	⊢ 𝐵 ∈ ℝ
lt.3	⊢ 𝐶 ∈ ℝ

Assertion

Ref	Expression
le2tri3i	⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴))

Proof of Theorem le2tri3i

Step	Hyp	Ref	Expression
1		lt.2	. . . . . 6 ⊢ 𝐵 ∈ ℝ
2		lt.3	. . . . . 6 ⊢ 𝐶 ∈ ℝ
3		lt.1	. . . . . 6 ⊢ 𝐴 ∈ ℝ
4	1, 2, 3	letri 11342	. . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 ≤ 𝐴)
5	3, 1	letri3i 11329	. . . . . 6 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))
6	5	biimpri 231	. . . . 5 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) → 𝐴 = 𝐵)
7	4, 6	sylan2 604	. . . 4 ⊢ ((𝐴 ≤ 𝐵 ∧ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) → 𝐴 = 𝐵)
8	7	3impb 1130	. . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐵)
9	2, 3, 1	letri 11342	. . . . . 6 ⊢ ((𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐶 ≤ 𝐵)
10	1, 2	letri3i 11329	. . . . . . 7 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
11	10	biimpri 231	. . . . . 6 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐵 = 𝐶)
12	9, 11	sylan2 604	. . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ (𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐵 = 𝐶)
13	12	3impb 1130	. . . 4 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐵 = 𝐶)
14	13	3comr 1141	. . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 = 𝐶)
15	3, 1, 2	letri 11342	. . . 4 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)
16	3, 2	letri3i 11329	. . . . . 6 ⊢ (𝐴 = 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴))
17	16	biimpri 231	. . . . 5 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐶)
18	17	eqcomd 2776	. . . 4 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴)
19	15, 18	stoic3 1804	. . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴)
20	8, 14, 19	3jca 1144	. 2 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴))
21	3	eqlei 11323	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ≤ 𝐵)
22	1	eqlei 11323	. . 3 ⊢ (𝐵 = 𝐶 → 𝐵 ≤ 𝐶)
23	2	eqlei 11323	. . 3 ⊢ (𝐶 = 𝐴 → 𝐶 ≤ 𝐴)
24	21, 22, 23	3anim123i 1167	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴) → (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴))
25	20, 24	impbii 212	1 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150   class class class wbr 5114  ℝcr 11102   ≤ cle 11247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736  ax-resscn 11160  ax-pre-lttri 11177  ax-pre-lttrn 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-nel 3072  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-po 5573  df-so 5574  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-er 8697  df-en 8947  df-dom 8948  df-sdom 8949  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252

This theorem is referenced by: (None)