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| Mirrors > Home > MPE Home > Th. List > elxrge0 | Structured version Visualization version GIF version | ||
| Description: Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.) |
| Ref | Expression |
|---|---|
| elxrge0 | ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-3an 1089 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞) ↔ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ 𝐴 ≤ +∞)) | |
| 2 | 0xr 11308 | . . 3 ⊢ 0 ∈ ℝ* | |
| 3 | pnfxr 11315 | . . 3 ⊢ +∞ ∈ ℝ* | |
| 4 | elicc1 13431 | . . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞))) | |
| 5 | 2, 3, 4 | mp2an 692 | . 2 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ +∞)) |
| 6 | pnfge 13172 | . . . 4 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) | |
| 7 | 6 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≤ +∞) |
| 8 | 7 | pm4.71i 559 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ↔ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ 𝐴 ≤ +∞)) |
| 9 | 1, 5, 8 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 ≤ cle 11296 [,]cicc 13390 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-icc 13394 |
| This theorem is referenced by: 0e0iccpnf 13499 ge0xaddcl 13502 ge0xmulcl 13503 xnn0xrge0 13546 xrge0subm 21425 psmetxrge0 24323 isxmet2d 24337 prdsdsf 24377 prdsxmetlem 24378 comet 24526 stdbdxmet 24528 xrge0gsumle 24855 xrge0tsms 24856 metdsf 24870 metds0 24872 metdstri 24873 metdsre 24875 metdseq0 24876 metdscnlem 24877 metnrmlem1a 24880 xrhmeo 24977 lebnumlem1 24993 xrge0f 25766 itg2const2 25776 itg2uba 25778 itg2mono 25788 itg2gt0 25795 itg2cnlem2 25797 itg2cn 25798 iblss 25840 itgle 25845 itgeqa 25849 ibladdlem 25855 iblabs 25864 iblabsr 25865 iblmulc2 25866 itgsplit 25871 bddmulibl 25874 bddiblnc 25877 xrge0addge 32761 xrge0infss 32764 xrge0addcld 32766 xrge0subcld 32767 xrge00 33017 xrge0tsmsd 33065 fldextrspundglemul 33729 esummono 34055 gsumesum 34060 esumsnf 34065 esumrnmpt2 34069 esumpmono 34080 hashf2 34085 measge0 34208 measle0 34209 measssd 34216 measunl 34217 omssubaddlem 34301 omssubadd 34302 carsgsigalem 34317 pmeasmono 34326 sibfinima 34341 prob01 34415 dstrvprob 34474 itg2addnclem 37678 ibladdnclem 37683 iblabsnc 37691 iblmulc2nc 37692 ftc1anclem4 37703 ftc1anclem5 37704 ftc1anclem6 37705 ftc1anclem7 37706 ftc1anclem8 37707 ftc1anc 37708 xrge0ge0 45358 rrxsphere 48669 |
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