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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 9723 | . 2 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴) | |
2 | xrlenlt 7971 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐴)) | |
3 | 2 | anidms 395 | . 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 2141 class class class wbr 3987 ℝ*cxr 7940 < clt 7941 ≤ cle 7942 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-pre-ltirr 7873 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-cnv 4617 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 |
This theorem is referenced by: xrleidd 9745 iccid 9869 ubioc1 9873 lbico1 9874 lbicc2 9928 ubicc2 9929 xrmaxifle 11196 |
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