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Mirrors > Home > ILE Home > Th. List > squeeze0 | GIF version |
Description: If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.) |
Ref | Expression |
---|---|
squeeze0 | ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7465 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
2 | 1 | 3ad2ant1 960 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → ¬ 𝐴 < 𝐴) |
3 | breq2 3815 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴)) | |
4 | breq2 3815 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴 < 𝑥 ↔ 𝐴 < 𝐴)) | |
5 | 3, 4 | imbi12d 232 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((0 < 𝑥 → 𝐴 < 𝑥) ↔ (0 < 𝐴 → 𝐴 < 𝐴))) |
6 | 5 | rspcva 2710 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → (0 < 𝐴 → 𝐴 < 𝐴)) |
7 | 6 | 3adant2 958 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → (0 < 𝐴 → 𝐴 < 𝐴)) |
8 | 2, 7 | mtod 622 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → ¬ 0 < 𝐴) |
9 | simp1 939 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 ∈ ℝ) | |
10 | 0red 7392 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 0 ∈ ℝ) | |
11 | 9, 10 | lenltd 7504 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → (𝐴 ≤ 0 ↔ ¬ 0 < 𝐴)) |
12 | 8, 11 | mpbird 165 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 ≤ 0) |
13 | simp2 940 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 0 ≤ 𝐴) | |
14 | 9, 10 | letri3d 7503 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → (𝐴 = 0 ↔ (𝐴 ≤ 0 ∧ 0 ≤ 𝐴))) |
15 | 12, 13, 14 | mpbir2and 886 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 = 0) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 ∀wral 2353 class class class wbr 3811 ℝcr 7252 0cc0 7253 < clt 7425 ≤ cle 7426 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-cnex 7339 ax-resscn 7340 ax-1re 7342 ax-addrcl 7345 ax-rnegex 7357 ax-pre-ltirr 7360 ax-pre-apti 7363 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-xp 4407 df-cnv 4409 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 |
This theorem is referenced by: (None) |
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