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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 13149 | . . 3 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ∈ ℝ*) | |
2 | xrge0neqmnf 13166 | . . 3 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞) | |
3 | xrnemnf 12835 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) | |
4 | 3 | biimpi 215 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
5 | 1, 2, 4 | syl2anc 583 | . 2 ⊢ (𝐴 ∈ (0[,]+∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
6 | 5 | orcanai 999 | 1 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ ¬ 𝐴 ∈ ℝ) → 𝐴 = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 843 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 (class class class)co 7268 ℝcr 10854 0cc0 10855 +∞cpnf 10990 -∞cmnf 10991 ℝ*cxr 10992 [,]cicc 13064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-addrcl 10916 ax-rnegex 10926 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-ov 7271 df-oprab 7272 df-mpo 7273 df-1st 7817 df-2nd 7818 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-icc 13068 |
This theorem is referenced by: voliune 32176 volfiniune 32177 omssubadd 32246 ismbl3 43481 |
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