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Mirrors > Home > ILE Home > Th. List > xrre3 | GIF version |
Description: A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.) |
Ref | Expression |
---|---|
xrre3 | ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnflt 9599 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵) | |
2 | 1 | adantl 275 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵) |
3 | mnfxr 7846 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
4 | 3 | a1i 9 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → -∞ ∈ ℝ*) |
5 | rexr 7835 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
6 | 5 | adantl 275 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ*) |
7 | simpl 108 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ*) | |
8 | xrltletr 9620 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝐴) → -∞ < 𝐴)) | |
9 | 4, 6, 7, 8 | syl3anc 1217 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝐴) → -∞ < 𝐴)) |
10 | 2, 9 | mpand 426 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ 𝐴 → -∞ < 𝐴)) |
11 | 10 | imp 123 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ 𝐵 ≤ 𝐴) → -∞ < 𝐴) |
12 | 11 | adantrr 471 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → -∞ < 𝐴) |
13 | simprr 522 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 < +∞) | |
14 | xrrebnd 9632 | . . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) | |
15 | 14 | ad2antrr 480 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
16 | 12, 13, 15 | mpbir2and 929 | 1 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1481 class class class wbr 3937 ℝcr 7643 +∞cpnf 7821 -∞cmnf 7822 ℝ*cxr 7823 < clt 7824 ≤ cle 7825 |
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 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 |
This theorem depends on definitions: df-bi 116 df-3or 964 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-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-po 4226 df-iso 4227 df-xp 4553 df-cnv 4555 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 |
This theorem is referenced by: (None) |
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