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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 6892 | . . . . 5 ⊢ (A ∈ ℝ → ¬ A < A) | |
2 | 1 | 3ad2ant1 924 | . . . 4 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → ¬ A < A) |
3 | breq2 3759 | . . . . . . 7 ⊢ (x = A → (0 < x ↔ 0 < A)) | |
4 | breq2 3759 | . . . . . . 7 ⊢ (x = A → (A < x ↔ A < A)) | |
5 | 3, 4 | imbi12d 223 | . . . . . 6 ⊢ (x = A → ((0 < x → A < x) ↔ (0 < A → A < A))) |
6 | 5 | rspcva 2648 | . . . . 5 ⊢ ((A ∈ ℝ ∧ ∀x ∈ ℝ (0 < x → A < x)) → (0 < A → A < A)) |
7 | 6 | 3adant2 922 | . . . 4 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → (0 < A → A < A)) |
8 | 2, 7 | mtod 588 | . . 3 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → ¬ 0 < A) |
9 | simp1 903 | . . . 4 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → A ∈ ℝ) | |
10 | 0red 6826 | . . . 4 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → 0 ∈ ℝ) | |
11 | 9, 10 | lenltd 6931 | . . 3 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → (A ≤ 0 ↔ ¬ 0 < A)) |
12 | 8, 11 | mpbird 156 | . 2 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → A ≤ 0) |
13 | simp2 904 | . 2 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → 0 ≤ A) | |
14 | 9, 10 | letri3d 6930 | . 2 ⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → (A = 0 ↔ (A ≤ 0 ∧ 0 ≤ A))) |
15 | 12, 13, 14 | mpbir2and 850 | 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 884 = wceq 1242 ∈ wcel 1390 ∀wral 2300 class class class wbr 3755 ℝcr 6710 0cc0 6711 < clt 6857 ≤ cle 6858 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-1re 6777 ax-addrcl 6780 ax-rnegex 6792 ax-pre-ltirr 6795 ax-pre-apti 6798 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-xp 4294 df-cnv 4296 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 |
This theorem is referenced by: (None) |
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