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| Description: Lemma for leordtval 23222. (Contributed by Mario Carneiro, 3-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| leordtval.1 | ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) | 
| Ref | Expression | 
|---|---|
| leordtvallem1 | ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | leordtval.1 | . 2 ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) | |
| 2 | iocssxr 13472 | . . . . . 6 ⊢ (𝑥(,]+∞) ⊆ ℝ* | |
| 3 | sseqin2 4222 | . . . . . 6 ⊢ ((𝑥(,]+∞) ⊆ ℝ* ↔ (ℝ* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞)) | |
| 4 | 2, 3 | mpbi 230 | . . . . 5 ⊢ (ℝ* ∩ (𝑥(,]+∞)) = (𝑥(,]+∞) | 
| 5 | simpl 482 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑥 ∈ ℝ*) | |
| 6 | pnfxr 11316 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
| 7 | elioc1 13430 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞))) | |
| 8 | 5, 6, 7 | sylancl 586 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞))) | 
| 9 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑦 ∈ ℝ*) | |
| 10 | pnfge 13173 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ* → 𝑦 ≤ +∞) | |
| 11 | 9, 10 | jccir 521 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞)) | 
| 12 | 11 | biantrurd 532 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ ((𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦))) | 
| 13 | 3anan32 1096 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞) ↔ ((𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞) ∧ 𝑥 < 𝑦)) | |
| 14 | 12, 13 | bitr4di 289 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ (𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞))) | 
| 15 | xrltnle 11329 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥)) | |
| 16 | 8, 14, 15 | 3bitr2d 307 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑦 ∈ (𝑥(,]+∞) ↔ ¬ 𝑦 ≤ 𝑥)) | 
| 17 | 16 | rabbi2dva 4225 | . . . . 5 ⊢ (𝑥 ∈ ℝ* → (ℝ* ∩ (𝑥(,]+∞)) = {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) | 
| 18 | 4, 17 | eqtr3id 2790 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (𝑥(,]+∞) = {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) | 
| 19 | 18 | mpteq2ia 5244 | . . 3 ⊢ (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) | 
| 20 | 19 | rneqi 5947 | . 2 ⊢ ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) | 
| 21 | 1, 20 | eqtri 2764 | 1 ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 {crab 3435 ∩ cin 3949 ⊆ wss 3950 class class class wbr 5142 ↦ cmpt 5224 ran crn 5685 (class class class)co 7432 +∞cpnf 11293 ℝ*cxr 11295 < clt 11296 ≤ cle 11297 (,]cioc 13389 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-ioc 13393 | 
| This theorem is referenced by: leordtval2 23221 leordtval 23222 | 
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