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Theorem ltne 7471
Description: 'Less than' implies not equal. See also ltap 8006 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
ltne ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵𝐴)

Proof of Theorem ltne
StepHypRef Expression
1 ltnr 7463 . . . 4 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
2 breq2 3815 . . . . 5 (𝐵 = 𝐴 → (𝐴 < 𝐵𝐴 < 𝐴))
32notbid 625 . . . 4 (𝐵 = 𝐴 → (¬ 𝐴 < 𝐵 ↔ ¬ 𝐴 < 𝐴))
41, 3syl5ibrcom 155 . . 3 (𝐴 ∈ ℝ → (𝐵 = 𝐴 → ¬ 𝐴 < 𝐵))
54necon2ad 2306 . 2 (𝐴 ∈ ℝ → (𝐴 < 𝐵𝐵𝐴))
65imp 122 1 ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102   = wceq 1285  wcel 1434  wne 2249   class class class wbr 3811  cr 7250   < clt 7423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-cnex 7337  ax-resscn 7338  ax-pre-ltirr 7358
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-xp 4405  df-pnf 7425  df-mnf 7426  df-ltxr 7428
This theorem is referenced by:  gtneii  7481  ltnei  7489  gtned  7498  gt0ne0  7806  lt0ne0  7807  gt0ne0d  7888  nngt1ne1  8348  zdceq  8716  qdceq  9545  coprm  10901  phibndlem  10970
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