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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 7846 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | pnfxr 7842 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | iccval 9733 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞[,]+∞) = {𝑥 ∈ ℝ* ∣ (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)}) | |
4 | 1, 2, 3 | mp2an 423 | . 2 ⊢ (-∞[,]+∞) = {𝑥 ∈ ℝ* ∣ (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)} |
5 | rabid2 2610 | . . 3 ⊢ (ℝ* = {𝑥 ∈ ℝ* ∣ (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)} ↔ ∀𝑥 ∈ ℝ* (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)) | |
6 | mnfle 9608 | . . . 4 ⊢ (𝑥 ∈ ℝ* → -∞ ≤ 𝑥) | |
7 | pnfge 9605 | . . . 4 ⊢ (𝑥 ∈ ℝ* → 𝑥 ≤ +∞) | |
8 | 6, 7 | jca 304 | . . 3 ⊢ (𝑥 ∈ ℝ* → (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)) |
9 | 5, 8 | mprgbir 2493 | . 2 ⊢ ℝ* = {𝑥 ∈ ℝ* ∣ (-∞ ≤ 𝑥 ∧ 𝑥 ≤ +∞)} |
10 | 4, 9 | eqtr4i 2164 | 1 ⊢ (-∞[,]+∞) = ℝ* |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 ∈ wcel 1481 {crab 2421 class class class wbr 3937 (class class class)co 5782 +∞cpnf 7821 -∞cmnf 7822 ℝ*cxr 7823 ≤ cle 7825 [,]cicc 9704 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-icc 9708 |
This theorem is referenced by: (None) |
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