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Mirrors > Home > MPE Home > Th. List > xrrebnd | Structured version Visualization version GIF version |
Description: An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.) |
Ref | Expression |
---|---|
xrrebnd | ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnflt 12506 | . . 3 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
2 | ltpnf 12503 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
3 | 1, 2 | jca 512 | . 2 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ∧ 𝐴 < +∞)) |
4 | nltpnft 12545 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) | |
5 | ngtmnft 12547 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) | |
6 | 4, 5 | orbi12d 912 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → ((𝐴 = +∞ ∨ 𝐴 = -∞) ↔ (¬ 𝐴 < +∞ ∨ ¬ -∞ < 𝐴))) |
7 | ianor 975 | . . . . . 6 ⊢ (¬ (-∞ < 𝐴 ∧ 𝐴 < +∞) ↔ (¬ -∞ < 𝐴 ∨ ¬ 𝐴 < +∞)) | |
8 | orcom 864 | . . . . . 6 ⊢ ((¬ -∞ < 𝐴 ∨ ¬ 𝐴 < +∞) ↔ (¬ 𝐴 < +∞ ∨ ¬ -∞ < 𝐴)) | |
9 | 7, 8 | bitr2i 277 | . . . . 5 ⊢ ((¬ 𝐴 < +∞ ∨ ¬ -∞ < 𝐴) ↔ ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞)) |
10 | 6, 9 | syl6bb 288 | . . . 4 ⊢ (𝐴 ∈ ℝ* → ((𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
11 | 10 | con2bid 356 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((-∞ < 𝐴 ∧ 𝐴 < +∞) ↔ ¬ (𝐴 = +∞ ∨ 𝐴 = -∞))) |
12 | elxr 12499 | . . . . 5 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
13 | 3orass 1082 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ (𝐴 ∈ ℝ ∨ (𝐴 = +∞ ∨ 𝐴 = -∞))) | |
14 | orcom 864 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∨ (𝐴 = +∞ ∨ 𝐴 = -∞)) ↔ ((𝐴 = +∞ ∨ 𝐴 = -∞) ∨ 𝐴 ∈ ℝ)) | |
15 | 13, 14 | bitri 276 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 = +∞ ∨ 𝐴 = -∞) ∨ 𝐴 ∈ ℝ)) |
16 | 12, 15 | sylbb 220 | . . . 4 ⊢ (𝐴 ∈ ℝ* → ((𝐴 = +∞ ∨ 𝐴 = -∞) ∨ 𝐴 ∈ ℝ)) |
17 | 16 | ord 858 | . . 3 ⊢ (𝐴 ∈ ℝ* → (¬ (𝐴 = +∞ ∨ 𝐴 = -∞) → 𝐴 ∈ ℝ)) |
18 | 11, 17 | sylbid 241 | . 2 ⊢ (𝐴 ∈ ℝ* → ((-∞ < 𝐴 ∧ 𝐴 < +∞) → 𝐴 ∈ ℝ)) |
19 | 3, 18 | impbid2 227 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 ∨ w3o 1078 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ℝcr 10524 +∞cpnf 10660 -∞cmnf 10661 ℝ*cxr 10662 < clt 10663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 |
This theorem is referenced by: xrre 12550 xrre2 12551 xrre3 12552 supxrre1 12711 elioc2 12787 elico2 12788 elicc2 12789 xblpnfps 22932 xblpnf 22933 isnghm3 23261 ovoliun 24033 ovolicopnf 24052 voliunlem3 24080 volsup 24084 itg2seq 24270 nmblore 28490 nmopre 29574 supxrgere 41477 supxrgelem 41481 supxrge 41482 suplesup 41483 infrpge 41495 limsupre 41798 |
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