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Mirrors > Home > ILE Home > Th. List > iccmax | 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 7935 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | pnfxr 7931 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | iccval 9825 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞[,]+∞) = {𝑥 ∈ ℝ* ∣ (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)}) | |
4 | 1, 2, 3 | mp2an 423 | . 2 ⊢ (-∞[,]+∞) = {𝑥 ∈ ℝ* ∣ (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)} |
5 | rabid2 2633 | . . 3 ⊢ (ℝ* = {𝑥 ∈ ℝ* ∣ (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)} ↔ ∀𝑥 ∈ ℝ* (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)) | |
6 | mnfle 9700 | . . . 4 ⊢ (𝑥 ∈ ℝ* → -∞ ≤ 𝑥) | |
7 | pnfge 9697 | . . . 4 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ +∞) | |
8 | 6, 7 | jca 304 | . . 3 ⊢ (𝑥 ∈ ℝ* → (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)) |
9 | 5, 8 | mprgbir 2515 | . 2 ⊢ ℝ* = {𝑥 ∈ ℝ* ∣ (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)} |
10 | 4, 9 | eqtr4i 2181 | 1 ⊢ (-∞[,]+∞) = ℝ* |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1335 ∈ wcel 2128 {crab 2439 class class class wbr 3966 (class class class)co 5825 +∞cpnf 7910 -∞cmnf 7911 ℝ*cxr 7912 ≤ cle 7914 [,]cicc 9796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-icc 9800 |
This theorem is referenced by: (None) |
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