Theorem le2tri3 4311

Description: Extended trichotomy law for 'less than or equal to'.

Hypotheses

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

Assertion

Ref	Expression
le2tri3	⊢ ((A ≤ B ∧ B ≤ C ∧ C ≤ A) ↔ (A = B ∧ B = C ∧ C = A))

Proof of Theorem le2tri3

Step	Hyp	Ref	Expression
1		lt.1	. . . . . . 7 ⊢ A ∈ ℝ
2		lt.2	. . . . . . 7 ⊢ B ∈ ℝ
3	1, 2	letri3 4297	. . . . . 6 ⊢ (A = B ↔ (A ≤ B ∧ B ≤ A))
4	3	biimpr 134	. . . . 5 ⊢ ((A ≤ B ∧ B ≤ A) → A = B)
5		lt.3	. . . . . 6 ⊢ C ∈ ℝ
6	2, 5, 1	letr 4310	. . . . 5 ⊢ ((B ≤ C ∧ C ≤ A) → B ≤ A)
7	4, 6	sylan2 346	. . . 4 ⊢ ((A ≤ B ∧ (B ≤ C ∧ C ≤ A)) → A = B)
8	7	3impb 610	. . 3 ⊢ ((A ≤ B ∧ B ≤ C ∧ C ≤ A) → A = B)
9	2, 5	letri3 4297	. . . . . . 7 ⊢ (B = C ↔ (B ≤ C ∧ C ≤ B))
10	9	biimpr 134	. . . . . 6 ⊢ ((B ≤ C ∧ C ≤ B) → B = C)
11	5, 1, 2	letr 4310	. . . . . 6 ⊢ ((C ≤ A ∧ A ≤ B) → C ≤ B)
12	10, 11	sylan2 346	. . . . 5 ⊢ ((B ≤ C ∧ (C ≤ A ∧ A ≤ B)) → B = C)
13	12	3impb 610	. . . 4 ⊢ ((B ≤ C ∧ C ≤ A ∧ A ≤ B) → B = C)
14	13	3comr 618	. . 3 ⊢ ((A ≤ B ∧ B ≤ C ∧ C ≤ A) → B = C)
15	1, 5	letri3 4297	. . . . . . 7 ⊢ (A = C ↔ (A ≤ C ∧ C ≤ A))
16	15	biimpr 134	. . . . . 6 ⊢ ((A ≤ C ∧ C ≤ A) → A = C)
17	16	cleqcomd 1106	. . . . 5 ⊢ ((A ≤ C ∧ C ≤ A) → C = A)
18	1, 2, 5	letr 4310	. . . . 5 ⊢ ((A ≤ B ∧ B ≤ C) → A ≤ C)
19	17, 18	sylan 343	. . . 4 ⊢ (((A ≤ B ∧ B ≤ C) ∧ C ≤ A) → C = A)
20	19	3impa 609	. . 3 ⊢ ((A ≤ B ∧ B ≤ C ∧ C ≤ A) → C = A)
21	8, 14, 20	3jca 604	. 2 ⊢ ((A ≤ B ∧ B ≤ C ∧ C ≤ A) → (A = B ∧ B = C ∧ C = A))
22	1, 2	eqle 4304	. . 3 ⊢ (A = B → A ≤ B)
23	2, 5	eqle 4304	. . 3 ⊢ (B = C → B ≤ C)
24	5, 1	eqle 4304	. . 3 ⊢ (C = A → C ≤ A)
25	22, 23, 24	im3an 605	. 2 ⊢ ((A = B ∧ B = C ∧ C = A) → (A ≤ B ∧ B ≤ C ∧ C ≤ A))
26	21, 25	impbi 139	1 ⊢ ((A ≤ B ∧ B ≤ C ∧ C ≤ A) ↔ (A = B ∧ B = C ∧ C = A))

Syntax hints:   ↔ wb 127   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092   class class class wbr 2054  ℝcr 4027   ≤ cle 4092

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-ltp 3884  df-enr 3960  df-nr 3961  df-ltr 3964  df-0r 3965  df-r 4038  df-lt 4041  df-le 4277

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