Theorem le2tri3i 11267

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 11266	. . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 ≤ 𝐴)
5	3, 1	letri3i 11253	. . . . . 6 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))
6	5	biimpri 228	. . . . 5 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) → 𝐴 = 𝐵)
7	4, 6	sylan2 594	. . . 4 ⊢ ((𝐴 ≤ 𝐵 ∧ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) → 𝐴 = 𝐵)
8	7	3impb 1115	. . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐵)
9	2, 3, 1	letri 11266	. . . . . 6 ⊢ ((𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐶 ≤ 𝐵)
10	1, 2	letri3i 11253	. . . . . . 7 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
11	10	biimpri 228	. . . . . 6 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐵 = 𝐶)
12	9, 11	sylan2 594	. . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ (𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐵 = 𝐶)
13	12	3impb 1115	. . . 4 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐵 = 𝐶)
14	13	3comr 1126	. . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 = 𝐶)
15	3, 1, 2	letri 11266	. . . 4 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)
16	3, 2	letri3i 11253	. . . . . 6 ⊢ (𝐴 = 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴))
17	16	biimpri 228	. . . . 5 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐶)
18	17	eqcomd 2743	. . . 4 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴)
19	15, 18	stoic3 1778	. . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴)
20	8, 14, 19	3jca 1129	. 2 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴))
21	3	eqlei 11247	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ≤ 𝐵)
22	1	eqlei 11247	. . 3 ⊢ (𝐵 = 𝐶 → 𝐵 ≤ 𝐶)
23	2	eqlei 11247	. . 3 ⊢ (𝐶 = 𝐴 → 𝐶 ≤ 𝐴)
24	21, 22, 23	3anim123i 1152	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴) → (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴))
25	20, 24	impbii 209	1 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5099  ℝcr 11029   ≤ cle 11171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-resscn 11087  ax-pre-lttri 11104  ax-pre-lttrn 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176

This theorem is referenced by: (None)