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Theorem le2tri3i 6923
 Description: Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)
Hypotheses
Ref Expression
lt.1 A
lt.2 B
lt.3 𝐶
Assertion
Ref Expression
le2tri3i ((AB B𝐶 𝐶A) ↔ (A = B B = 𝐶 𝐶 = A))

Proof of Theorem le2tri3i
StepHypRef Expression
1 lt.2 . . . . . 6 B
2 lt.3 . . . . . 6 𝐶
3 lt.1 . . . . . 6 A
41, 2, 3letri 6922 . . . . 5 ((B𝐶 𝐶A) → BA)
53, 1letri3i 6913 . . . . . 6 (A = B ↔ (AB BA))
65biimpri 124 . . . . 5 ((AB BA) → A = B)
74, 6sylan2 270 . . . 4 ((AB (B𝐶 𝐶A)) → A = B)
873impb 1099 . . 3 ((AB B𝐶 𝐶A) → A = B)
92, 3, 1letri 6922 . . . . . 6 ((𝐶A AB) → 𝐶B)
101, 2letri3i 6913 . . . . . . 7 (B = 𝐶 ↔ (B𝐶 𝐶B))
1110biimpri 124 . . . . . 6 ((B𝐶 𝐶B) → B = 𝐶)
129, 11sylan2 270 . . . . 5 ((B𝐶 (𝐶A AB)) → B = 𝐶)
13123impb 1099 . . . 4 ((B𝐶 𝐶A AB) → B = 𝐶)
14133comr 1111 . . 3 ((AB B𝐶 𝐶A) → B = 𝐶)
153, 1, 2letri 6922 . . . . 5 ((AB B𝐶) → A𝐶)
163, 2letri3i 6913 . . . . . . 7 (A = 𝐶 ↔ (A𝐶 𝐶A))
1716biimpri 124 . . . . . 6 ((A𝐶 𝐶A) → A = 𝐶)
1817eqcomd 2042 . . . . 5 ((A𝐶 𝐶A) → 𝐶 = A)
1915, 18sylan 267 . . . 4 (((AB B𝐶) 𝐶A) → 𝐶 = A)
20193impa 1098 . . 3 ((AB B𝐶 𝐶A) → 𝐶 = A)
218, 14, 203jca 1083 . 2 ((AB B𝐶 𝐶A) → (A = B B = 𝐶 𝐶 = A))
223eqlei 6908 . . 3 (A = BAB)
231eqlei 6908 . . 3 (B = 𝐶B𝐶)
242eqlei 6908 . . 3 (𝐶 = A𝐶A)
2522, 23, 243anim123i 1088 . 2 ((A = B B = 𝐶 𝐶 = A) → (AB B𝐶 𝐶A))
2621, 25impbii 117 1 ((AB B𝐶 𝐶A) ↔ (A = B B = 𝐶 𝐶 = A))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390   class class class wbr 3755  ℝcr 6710   ≤ cle 6858 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-pre-ltirr 6795  ax-pre-ltwlin 6796  ax-pre-apti 6798 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863 This theorem is referenced by: (None)
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