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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 9791 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) | |
2 | 1 | biimpd 144 | . . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ → ¬ -∞ < 𝐴)) |
3 | 2 | necon2ad 2404 | . 2 ⊢ (𝐴 ∈ ℝ* → (-∞ < 𝐴 → 𝐴 ≠ -∞)) |
4 | mnflt 9757 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
5 | 4 | adantl 277 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ 𝐴 ∈ ℝ) → -∞ < 𝐴) |
6 | mnfltpnf 9759 | . . . . . 6 ⊢ -∞ < +∞ | |
7 | breq2 4004 | . . . . . 6 ⊢ (𝐴 = +∞ → (-∞ < 𝐴 ↔ -∞ < +∞)) | |
8 | 6, 7 | mpbiri 168 | . . . . 5 ⊢ (𝐴 = +∞ → -∞ < 𝐴) |
9 | 8 | adantl 277 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ 𝐴 = +∞) → -∞ < 𝐴) |
10 | simpr 110 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ 𝐴 = -∞) → 𝐴 = -∞) | |
11 | simplr 528 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ 𝐴 = -∞) → 𝐴 ≠ -∞) | |
12 | 10, 11 | pm2.21ddne 2430 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ 𝐴 = -∞) → -∞ < 𝐴) |
13 | elxr 9750 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
14 | 13 | biimpi 120 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
15 | 14 | adantr 276 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) |
16 | 5, 9, 12, 15 | mpjao3dan 1307 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → -∞ < 𝐴) |
17 | 16 | ex 115 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≠ -∞ → -∞ < 𝐴)) |
18 | 3, 17 | impbid 129 | 1 ⊢ (𝐴 ∈ ℝ* → (-∞ < 𝐴 ↔ 𝐴 ≠ -∞)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ 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: xlt2add 9854 xrmaxadd 11240 xblpnfps 13531 xblpnf 13532 |
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