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| Mirrors > Home > ILE Home > Th. List > xrltled | GIF version | ||
| Description: 'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle 9933. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| xrltled.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
| xrltled.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
| xrltled.altb | ⊢ (𝜑 → 𝐴 < 𝐵) |
| Ref | Expression |
|---|---|
| xrltled | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrltled.altb | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
| 2 | xrltled.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
| 3 | xrltled.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
| 4 | xrltle 9933 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | |
| 5 | 2, 3, 4 | syl2anc 411 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
| 6 | 1, 5 | mpd 13 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∈ wcel 2177 class class class wbr 4048 ℝ*cxr 8119 < clt 8120 ≤ cle 8121 |
| 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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4167 ax-pow 4223 ax-pr 4258 ax-un 4485 ax-setind 4590 ax-cnex 8029 ax-resscn 8030 ax-pre-ltirr 8050 ax-pre-lttrn 8052 |
| This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-rab 2494 df-v 2775 df-dif 3170 df-un 3172 df-in 3174 df-ss 3181 df-pw 3620 df-sn 3641 df-pr 3642 df-op 3644 df-uni 3854 df-br 4049 df-opab 4111 df-xp 4686 df-cnv 4688 df-pnf 8122 df-mnf 8123 df-xr 8124 df-ltxr 8125 df-le 8126 |
| This theorem is referenced by: xrmaxadd 11622 xrbdtri 11637 pcadd2 12714 xblss2ps 14926 xblss2 14927 blhalf 14930 blssps 14949 blss 14950 bdmopn 15026 tgqioo 15077 |
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