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Mirrors > Home > ILE Home > Th. List > ltne | GIF version |
Description: 'Less than' implies not equal. See also ltap 8388 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
ltne | ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7834 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
2 | breq2 3928 | . . . . 5 ⊢ (𝐵 = 𝐴 → (𝐴 < 𝐵 ↔ 𝐴 < 𝐴)) | |
3 | 2 | notbid 656 | . . . 4 ⊢ (𝐵 = 𝐴 → (¬ 𝐴 < 𝐵 ↔ ¬ 𝐴 < 𝐴)) |
4 | 1, 3 | syl5ibrcom 156 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐵 = 𝐴 → ¬ 𝐴 < 𝐵)) |
5 | 4 | necon2ad 2363 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 𝐵 → 𝐵 ≠ 𝐴)) |
6 | 5 | imp 123 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ≠ wne 2306 class class class wbr 3924 ℝcr 7612 < clt 7793 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-pnf 7795 df-mnf 7796 df-ltxr 7798 |
This theorem is referenced by: gtneii 7852 ltnei 7860 gtned 7869 gt0ne0 8182 lt0ne0 8183 gt0ne0d 8267 nngt1ne1 8748 zdceq 9119 qdceq 10017 coprm 11811 phibndlem 11881 |
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