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Mirrors > Home > MPE Home > Th. List > iccmax | Structured version Visualization version GIF version |
Description: The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.) |
Ref | Expression |
---|---|
iccmax | ⊢ (-∞[,]+∞) = ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 10692 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | pnfxr 10689 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | iccval 12771 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞[,]+∞) = {𝑥 ∈ ℝ* ∣ (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)}) | |
4 | 1, 2, 3 | mp2an 690 | . 2 ⊢ (-∞[,]+∞) = {𝑥 ∈ ℝ* ∣ (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)} |
5 | rabid2 3382 | . . 3 ⊢ (ℝ* = {𝑥 ∈ ℝ* ∣ (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)} ↔ ∀𝑥 ∈ ℝ* (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)) | |
6 | mnfle 12523 | . . . 4 ⊢ (𝑥 ∈ ℝ* → -∞ ≤ 𝑥) | |
7 | pnfge 12519 | . . . 4 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ +∞) | |
8 | 6, 7 | jca 514 | . . 3 ⊢ (𝑥 ∈ ℝ* → (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)) |
9 | 5, 8 | mprgbir 3153 | . 2 ⊢ ℝ* = {𝑥 ∈ ℝ* ∣ (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)} |
10 | 4, 9 | eqtr4i 2847 | 1 ⊢ (-∞[,]+∞) = ℝ* |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 class class class wbr 5059 (class class class)co 7150 +∞cpnf 10666 -∞cmnf 10667 ℝ*cxr 10668 ≤ cle 10670 [,]cicc 12735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-icc 12739 |
This theorem is referenced by: leordtval2 21814 lecldbas 21821 |
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