Theorem le2tri3i 10772

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 10771	. . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 ≤ 𝐴)
5	3, 1	letri3i 10758	. . . . . 6 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))
6	5	biimpri 230	. . . . 5 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) → 𝐴 = 𝐵)
7	4, 6	sylan2 594	. . . 4 ⊢ ((𝐴 ≤ 𝐵 ∧ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) → 𝐴 = 𝐵)
8	7	3impb 1111	. . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐵)
9	2, 3, 1	letri 10771	. . . . . 6 ⊢ ((𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐶 ≤ 𝐵)
10	1, 2	letri3i 10758	. . . . . . 7 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
11	10	biimpri 230	. . . . . 6 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐵 = 𝐶)
12	9, 11	sylan2 594	. . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ (𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐵 = 𝐶)
13	12	3impb 1111	. . . 4 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐵 = 𝐶)
14	13	3comr 1121	. . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 = 𝐶)
15	3, 1, 2	letri 10771	. . . 4 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)
16	3, 2	letri3i 10758	. . . . . 6 ⊢ (𝐴 = 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴))
17	16	biimpri 230	. . . . 5 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐶)
18	17	eqcomd 2829	. . . 4 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴)
19	15, 18	stoic3 1777	. . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴)
20	8, 14, 19	3jca 1124	. 2 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴))
21	3	eqlei 10752	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ≤ 𝐵)
22	1	eqlei 10752	. . 3 ⊢ (𝐵 = 𝐶 → 𝐵 ≤ 𝐶)
23	2	eqlei 10752	. . 3 ⊢ (𝐶 = 𝐴 → 𝐶 ≤ 𝐴)
24	21, 22, 23	3anim123i 1147	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴) → (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴))
25	20, 24	impbii 211	1 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   class class class wbr 5068  ℝcr 10538   ≤ cle 10678

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-pow 5268  ax-pr 5332  ax-un 7463  ax-resscn 10596  ax-pre-lttri 10613  ax-pre-lttrn 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  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-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-po 5476  df-so 5477  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683

This theorem is referenced by: (None)