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Theorem le2tri3i 8080
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 ((𝐴𝐵𝐵𝐶𝐶𝐴) ↔ (𝐴 = 𝐵𝐵 = 𝐶𝐶 = 𝐴))

Proof of Theorem le2tri3i
StepHypRef Expression
1 lt.2 . . . . . 6 𝐵 ∈ ℝ
2 lt.3 . . . . . 6 𝐶 ∈ ℝ
3 lt.1 . . . . . 6 𝐴 ∈ ℝ
41, 2, 3letri 8079 . . . . 5 ((𝐵𝐶𝐶𝐴) → 𝐵𝐴)
53, 1letri3i 8070 . . . . . 6 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
65biimpri 133 . . . . 5 ((𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
74, 6sylan2 286 . . . 4 ((𝐴𝐵 ∧ (𝐵𝐶𝐶𝐴)) → 𝐴 = 𝐵)
873impb 1200 . . 3 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐴 = 𝐵)
92, 3, 1letri 8079 . . . . . 6 ((𝐶𝐴𝐴𝐵) → 𝐶𝐵)
101, 2letri3i 8070 . . . . . . 7 (𝐵 = 𝐶 ↔ (𝐵𝐶𝐶𝐵))
1110biimpri 133 . . . . . 6 ((𝐵𝐶𝐶𝐵) → 𝐵 = 𝐶)
129, 11sylan2 286 . . . . 5 ((𝐵𝐶 ∧ (𝐶𝐴𝐴𝐵)) → 𝐵 = 𝐶)
13123impb 1200 . . . 4 ((𝐵𝐶𝐶𝐴𝐴𝐵) → 𝐵 = 𝐶)
14133comr 1212 . . 3 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐵 = 𝐶)
153, 1, 2letri 8079 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
163, 2letri3i 8070 . . . . . . 7 (𝐴 = 𝐶 ↔ (𝐴𝐶𝐶𝐴))
1716biimpri 133 . . . . . 6 ((𝐴𝐶𝐶𝐴) → 𝐴 = 𝐶)
1817eqcomd 2193 . . . . 5 ((𝐴𝐶𝐶𝐴) → 𝐶 = 𝐴)
1915, 18sylan 283 . . . 4 (((𝐴𝐵𝐵𝐶) ∧ 𝐶𝐴) → 𝐶 = 𝐴)
20193impa 1195 . . 3 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶 = 𝐴)
218, 14, 203jca 1178 . 2 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵𝐵 = 𝐶𝐶 = 𝐴))
223eqlei 8065 . . 3 (𝐴 = 𝐵𝐴𝐵)
231eqlei 8065 . . 3 (𝐵 = 𝐶𝐵𝐶)
242eqlei 8065 . . 3 (𝐶 = 𝐴𝐶𝐴)
2522, 23, 243anim123i 1185 . 2 ((𝐴 = 𝐵𝐵 = 𝐶𝐶 = 𝐴) → (𝐴𝐵𝐵𝐶𝐶𝐴))
2621, 25impbii 126 1 ((𝐴𝐵𝐵𝐶𝐶𝐴) ↔ (𝐴 = 𝐵𝐵 = 𝐶𝐶 = 𝐴))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  w3a 979   = wceq 1363  wcel 2158   class class class wbr 4015  cr 7824  cle 8007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-apti 7940
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-cnv 4646  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012
This theorem is referenced by: (None)
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