![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nmnfgt | GIF version |
Description: An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.) |
Ref | Expression |
---|---|
nmnfgt | ⊢ (𝐴 ∈ ℝ* → (-∞ < 𝐴 ↔ 𝐴 ≠ -∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngtmnft 9441 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) | |
2 | 1 | biimpd 143 | . . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ → ¬ -∞ < 𝐴)) |
3 | 2 | necon2ad 2324 | . 2 ⊢ (𝐴 ∈ ℝ* → (-∞ < 𝐴 → 𝐴 ≠ -∞)) |
4 | mnflt 9410 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
5 | 4 | adantl 273 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ 𝐴 ∈ ℝ) → -∞ < 𝐴) |
6 | mnfltpnf 9412 | . . . . . 6 ⊢ -∞ < +∞ | |
7 | breq2 3879 | . . . . . 6 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞)) | |
8 | 6, 7 | mpbiri 167 | . . . . 5 ⊢ (𝐴 = +∞ → -∞ < 𝐴) |
9 | 8 | adantl 273 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ 𝐴 = +∞) → -∞ < 𝐴) |
10 | simpr 109 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ 𝐴 = -∞) → 𝐴 = -∞) | |
11 | simplr 500 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ 𝐴 = -∞) → 𝐴 ≠ -∞) | |
12 | 10, 11 | pm2.21ddne 2350 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ 𝐴 = -∞) → -∞ < 𝐴) |
13 | elxr 9404 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
14 | 13 | biimpi 119 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
15 | 14 | adantr 272 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
16 | 5, 9, 12, 15 | mpjao3dan 1253 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → -∞ < 𝐴) |
17 | 16 | ex 114 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≠ -∞ → -∞ < 𝐴)) |
18 | 3, 17 | impbid 128 | 1 ⊢ (𝐴 ∈ ℝ* → (-∞ < 𝐴 ↔ 𝐴 ≠ -∞)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ w3o 929 = wceq 1299 ∈ wcel 1448 ≠ wne 2267 class class class wbr 3875 ℝcr 7499 +∞cpnf 7669 -∞cmnf 7670 ℝ*cxr 7671 < clt 7672 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-pre-ltirr 7607 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-xp 4483 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 |
This theorem is referenced by: xlt2add 9504 xrmaxadd 10869 xblpnfps 12326 xblpnf 12327 |
Copyright terms: Public domain | W3C validator |