Theorem le2tri3i 11271

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 11270	. . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 ≤ 𝐴)
5	3, 1	letri3i 11257	. . . . . 6 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))
6	5	biimpri 230	. . . . 5 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) → 𝐴 = 𝐵)
7	4, 6	sylan2 600	. . . 4 ⊢ ((𝐴 ≤ 𝐵 ∧ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) → 𝐴 = 𝐵)
8	7	3impb 1121	. . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐵)
9	2, 3, 1	letri 11270	. . . . . 6 ⊢ ((𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐶 ≤ 𝐵)
10	1, 2	letri3i 11257	. . . . . . 7 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
11	10	biimpri 230	. . . . . 6 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐵 = 𝐶)
12	9, 11	sylan2 600	. . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ (𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐵 = 𝐶)
13	12	3impb 1121	. . . 4 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐵 = 𝐶)
14	13	3comr 1132	. . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 = 𝐶)
15	3, 1, 2	letri 11270	. . . 4 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)
16	3, 2	letri3i 11257	. . . . . 6 ⊢ (𝐴 = 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴))
17	16	biimpri 230	. . . . 5 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐶)
18	17	eqcomd 2747	. . . 4 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴)
19	15, 18	stoic3 1784	. . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴)
20	8, 14, 19	3jca 1135	. 2 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴))
21	3	eqlei 11251	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ≤ 𝐵)
22	1	eqlei 11251	. . 3 ⊢ (𝐵 = 𝐶 → 𝐵 ≤ 𝐶)
23	2	eqlei 11251	. . 3 ⊢ (𝐶 = 𝐴 → 𝐶 ≤ 𝐴)
24	21, 22, 23	3anim123i 1158	. 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 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   class class class wbr 5075  ℝcr 11032   ≤ cle 11175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-resscn 11090  ax-pre-lttri 11107  ax-pre-lttrn 11108

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-po 5529  df-so 5530  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180

This theorem is referenced by: (None)