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Mirrors > Home > MPE Home > Th. List > squeeze0 | Structured version Visualization version 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 | 0re 10328 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | leloe 10412 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴))) | |
3 | 1, 2 | mpan 682 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴))) |
4 | breq2 4845 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴)) | |
5 | breq2 4845 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴 < 𝑥 ↔ 𝐴 < 𝐴)) | |
6 | 4, 5 | imbi12d 336 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((0 < 𝑥 → 𝐴 < 𝑥) ↔ (0 < 𝐴 → 𝐴 < 𝐴))) |
7 | 6 | rspcv 3491 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥) → (0 < 𝐴 → 𝐴 < 𝐴))) |
8 | ltnr 10420 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
9 | 8 | pm2.21d 119 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 < 𝐴 → 𝐴 = 0)) |
10 | 9 | com12 32 | . . . . . . 7 ⊢ (𝐴 < 𝐴 → (𝐴 ∈ ℝ → 𝐴 = 0)) |
11 | 10 | imim2i 16 | . . . . . 6 ⊢ ((0 < 𝐴 → 𝐴 < 𝐴) → (0 < 𝐴 → (𝐴 ∈ ℝ → 𝐴 = 0))) |
12 | 11 | com13 88 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 → ((0 < 𝐴 → 𝐴 < 𝐴) → 𝐴 = 0))) |
13 | 7, 12 | syl5d 73 | . . . 4 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 → (∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥) → 𝐴 = 0))) |
14 | ax-1 6 | . . . . . 6 ⊢ (𝐴 = 0 → (∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥) → 𝐴 = 0)) | |
15 | 14 | eqcoms 2805 | . . . . 5 ⊢ (0 = 𝐴 → (∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥) → 𝐴 = 0)) |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℝ → (0 = 𝐴 → (∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥) → 𝐴 = 0))) |
17 | 13, 16 | jaod 886 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 < 𝐴 ∨ 0 = 𝐴) → (∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥) → 𝐴 = 0))) |
18 | 3, 17 | sylbid 232 | . 2 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 → (∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥) → 𝐴 = 0))) |
19 | 18 | 3imp 1138 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∨ wo 874 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ∀wral 3087 class class class wbr 4841 ℝcr 10221 0cc0 10222 < clt 10361 ≤ cle 10362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-resscn 10279 ax-1cn 10280 ax-addrcl 10283 ax-rnegex 10293 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-po 5231 df-so 5232 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 |
This theorem is referenced by: (None) |
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