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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 9459 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴) | |
2 | breq2 3899 | . . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐴 < 𝐵 ↔ 𝐴 < 𝐴)) | |
3 | 2 | notbid 639 | . . . . 5 ⊢ (𝐵 = 𝐴 → (¬ 𝐴 < 𝐵 ↔ ¬ 𝐴 < 𝐴)) |
4 | 1, 3 | syl5ibrcom 156 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐵 = 𝐴 → ¬ 𝐴 < 𝐵)) |
5 | 4 | necon2ad 2339 | . . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 < 𝐵 → 𝐵 ≠ 𝐴)) |
6 | 5 | imp 123 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
7 | 6 | 3adant2 983 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 945 = wceq 1314 ∈ wcel 1463 ≠ wne 2282 class class class wbr 3895 ℝ*cxr 7723 < clt 7724 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 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 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-pre-ltirr 7657 |
This theorem depends on definitions: df-bi 116 df-3or 946 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 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-xp 4505 df-pnf 7726 df-mnf 7727 df-xr 7728 df-ltxr 7729 |
This theorem is referenced by: (None) |
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