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Mirrors > Home > ILE Home > Th. List > ngtmnft | GIF version |
Description: An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.) |
Ref | Expression |
---|---|
ngtmnft | ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9750 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | renemnf 7983 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
3 | 2 | neneqd 2368 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = -∞) |
4 | mnflt 9757 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
5 | notnot 629 | . . . . 5 ⊢ (-∞ < 𝐴 → ¬ ¬ -∞ < 𝐴) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ ¬ -∞ < 𝐴) |
7 | 3, 6 | 2falsed 702 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
8 | pnfnemnf 7989 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
9 | neeq1 2360 | . . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ≠ -∞ ↔ +∞ ≠ -∞)) | |
10 | 8, 9 | mpbiri 168 | . . . . 5 ⊢ (𝐴 = +∞ → 𝐴 ≠ -∞) |
11 | 10 | neneqd 2368 | . . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 = -∞) |
12 | mnfltpnf 9759 | . . . . . . 7 ⊢ -∞ < +∞ | |
13 | breq2 4004 | . . . . . . 7 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞)) | |
14 | 12, 13 | mpbiri 168 | . . . . . 6 ⊢ (𝐴 = +∞ → -∞ < 𝐴) |
15 | 14 | necon3bi 2397 | . . . . 5 ⊢ (¬ -∞ < 𝐴 → 𝐴 ≠ +∞) |
16 | 15 | necon2bi 2402 | . . . 4 ⊢ (𝐴 = +∞ → ¬ ¬ -∞ < 𝐴) |
17 | 11, 16 | 2falsed 702 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
18 | id 19 | . . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
19 | mnfxr 7991 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
20 | xrltnr 9753 | . . . . . 6 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ ¬ -∞ < -∞ |
22 | breq2 4004 | . . . . 5 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞)) | |
23 | 21, 22 | mtbiri 675 | . . . 4 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴) |
24 | 18, 23 | 2thd 175 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
25 | 7, 17, 24 | 3jaoi 1303 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
26 | 1, 25 | sylbi 121 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 class class class wbr 4000 ℝcr 7788 +∞cpnf 7966 -∞cmnf 7967 ℝ*cxr 7968 < clt 7969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-pre-ltirr 7901 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4628 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 |
This theorem is referenced by: nmnfgt 9792 ge0nemnf 9798 xleaddadd 9861 |
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