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Theorem ltne 7045
Description: 'Less than' implies not equal. See also ltap 7560 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 7037 . . . 4 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
2 breq2 3764 . . . . 5 (𝐵 = 𝐴 → (𝐴 < 𝐵𝐴 < 𝐴))
32notbid 592 . . . 4 (𝐵 = 𝐴 → (¬ 𝐴 < 𝐵 ↔ ¬ 𝐴 < 𝐴))
41, 3syl5ibrcom 146 . . 3 (𝐴 ∈ ℝ → (𝐵 = 𝐴 → ¬ 𝐴 < 𝐵))
54necon2ad 2259 . 2 (𝐴 ∈ ℝ → (𝐴 < 𝐵𝐵𝐴))
65imp 115 1 ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97   = wceq 1243  wcel 1393  wne 2204   class class class wbr 3760  cr 6831   < clt 7002
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3871  ax-pow 3923  ax-pr 3940  ax-un 4141  ax-setind 4230  ax-cnex 6918  ax-resscn 6919  ax-pre-ltirr 6939
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3577  df-br 3761  df-opab 3815  df-xp 4312  df-pnf 7004  df-mnf 7005  df-ltxr 7007
This theorem is referenced by:  gtneii  7055  ltnei  7063  gtned  7072  gt0ne0  7362  lt0ne0  7363  gt0ne0d  7444  nngt1ne1  7886  zdceq  8250
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