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Mirrors > Home > MPE Home > Th. List > xrge0nre | Structured version Visualization version GIF version |
Description: An extended real which is not a real is plus infinity. (Contributed by Thierry Arnoux, 16-Oct-2017.) |
Ref | Expression |
---|---|
xrge0nre | ⊢ ((𝐴 ∈ (0[,]+∞) ∧ ¬ 𝐴 ∈ ℝ) → 𝐴 = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliccxr 12638 | . . 3 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ∈ ℝ*) | |
2 | xrge0neqmnf 12655 | . . 3 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞) | |
3 | xrnemnf 12328 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) | |
4 | 3 | biimpi 208 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
5 | 1, 2, 4 | syl2anc 576 | . 2 ⊢ (𝐴 ∈ (0[,]+∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
6 | 5 | orcanai 986 | 1 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ ¬ 𝐴 ∈ ℝ) → 𝐴 = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∨ wo 834 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 (class class class)co 6975 ℝcr 10333 0cc0 10334 +∞cpnf 10470 -∞cmnf 10471 ℝ*cxr 10472 [,]cicc 12556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-addrcl 10395 ax-rnegex 10405 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-po 5323 df-so 5324 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-ov 6978 df-oprab 6979 df-mpo 6980 df-1st 7500 df-2nd 7501 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-icc 12560 |
This theorem is referenced by: voliune 31166 volfiniune 31167 omssubadd 31236 ismbl3 41732 |
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