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Theorem ltne 7813
Description: 'Less than' implies not equal. See also ltap 8357 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 7805 . . . 4 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
2 breq2 3901 . . . . 5 (𝐵 = 𝐴 → (𝐴 < 𝐵𝐴 < 𝐴))
32notbid 639 . . . 4 (𝐵 = 𝐴 → (¬ 𝐴 < 𝐵 ↔ ¬ 𝐴 < 𝐴))
41, 3syl5ibrcom 156 . . 3 (𝐴 ∈ ℝ → (𝐵 = 𝐴 → ¬ 𝐴 < 𝐵))
54necon2ad 2340 . 2 (𝐴 ∈ ℝ → (𝐴 < 𝐵𝐵𝐴))
65imp 123 1 ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1314  wcel 1463  wne 2283   class class class wbr 3897  cr 7583   < clt 7764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-pre-ltirr 7696
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-pnf 7766  df-mnf 7767  df-ltxr 7769
This theorem is referenced by:  gtneii  7823  ltnei  7831  gtned  7840  gt0ne0  8153  lt0ne0  8154  gt0ne0d  8238  nngt1ne1  8712  zdceq  9077  qdceq  9964  coprm  11718  phibndlem  11787
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