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Mirrors > Home > ILE Home > Th. List > xrltne | GIF version |
Description: 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.) |
Ref | Expression |
---|---|
xrltne | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltnr 9848 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴) | |
2 | breq2 4034 | . . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐴 < 𝐵 ↔ 𝐴 < 𝐴)) | |
3 | 2 | notbid 668 | . . . . 5 ⊢ (𝐵 = 𝐴 → (¬ 𝐴 < 𝐵 ↔ ¬ 𝐴 < 𝐴)) |
4 | 1, 3 | syl5ibrcom 157 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐵 = 𝐴 → ¬ 𝐴 < 𝐵)) |
5 | 4 | necon2ad 2421 | . . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 < 𝐵 → 𝐵 ≠ 𝐴)) |
6 | 5 | imp 124 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
7 | 6 | 3adant2 1018 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 980 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 class class class wbr 4030 ℝ*cxr 8055 < clt 8056 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-pre-ltirr 7986 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-xp 4666 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 |
This theorem is referenced by: (None) |
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