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Mirrors > Home > ILE Home > Th. List > xrlenlt | GIF version |
Description: 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
xrlenlt | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3765 | . . 3 ⊢ (𝐴 ≤ 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ≤ ) | |
2 | opelxpi 4376 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 〈𝐴, 𝐵〉 ∈ (ℝ* × ℝ*)) | |
3 | df-le 7066 | . . . . . . 7 ⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < ) | |
4 | 3 | eleq2i 2104 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ ≤ ↔ 〈𝐴, 𝐵〉 ∈ ((ℝ* × ℝ*) ∖ ◡ < )) |
5 | eldif 2927 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ ((ℝ* × ℝ*) ∖ ◡ < ) ↔ (〈𝐴, 𝐵〉 ∈ (ℝ* × ℝ*) ∧ ¬ 〈𝐴, 𝐵〉 ∈ ◡ < )) | |
6 | 4, 5 | bitri 173 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ ≤ ↔ (〈𝐴, 𝐵〉 ∈ (ℝ* × ℝ*) ∧ ¬ 〈𝐴, 𝐵〉 ∈ ◡ < )) |
7 | 6 | baib 828 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (ℝ* × ℝ*) → (〈𝐴, 𝐵〉 ∈ ≤ ↔ ¬ 〈𝐴, 𝐵〉 ∈ ◡ < )) |
8 | 2, 7 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (〈𝐴, 𝐵〉 ∈ ≤ ↔ ¬ 〈𝐴, 𝐵〉 ∈ ◡ < )) |
9 | 1, 8 | syl5bb 181 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 〈𝐴, 𝐵〉 ∈ ◡ < )) |
10 | opelcnvg 4515 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (〈𝐴, 𝐵〉 ∈ ◡ < ↔ 〈𝐵, 𝐴〉 ∈ < )) | |
11 | df-br 3765 | . . . 4 ⊢ (𝐵 < 𝐴 ↔ 〈𝐵, 𝐴〉 ∈ < ) | |
12 | 10, 11 | syl6rbbr 188 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 < 𝐴 ↔ 〈𝐴, 𝐵〉 ∈ ◡ < )) |
13 | 12 | notbid 592 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (¬ 𝐵 < 𝐴 ↔ ¬ 〈𝐴, 𝐵〉 ∈ ◡ < )) |
14 | 9, 13 | bitr4d 180 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1393 ∖ cdif 2914 〈cop 3378 class class class wbr 3764 × cxp 4343 ◡ccnv 4344 ℝ*cxr 7059 < clt 7060 ≤ cle 7061 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-le 7066 |
This theorem is referenced by: lenlt 7094 pnfge 8710 mnfle 8713 xrltle 8719 xrleid 8720 xrletri3 8721 xrlelttr 8722 xrltletr 8723 xrletr 8724 xleneg 8750 iccid 8794 icc0r 8795 icodisj 8860 ioodisj 8861 |
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