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Mirrors > Home > ILE Home > Th. List > xrleid | GIF version |
Description: 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.) |
Ref | Expression |
---|---|
xrleid | ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltnr 9534 | . 2 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴) | |
2 | xrlenlt 7797 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐴)) | |
3 | 2 | anidms 394 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐴)) |
4 | 1, 3 | mpbird 166 | 1 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∈ wcel 1465 class class class wbr 3899 ℝ*cxr 7767 < clt 7768 ≤ cle 7769 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 |
This theorem is referenced by: xrleidd 9555 iccid 9676 ubioc1 9680 lbico1 9681 lbicc2 9735 ubicc2 9736 xrmaxifle 10983 |
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