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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 10643 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | leloe 10727 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴))) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴))) |
4 | breq2 5070 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴)) | |
5 | breq2 5070 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴 < 𝑥 ↔ 𝐴 < 𝐴)) | |
6 | 4, 5 | imbi12d 347 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((0 < 𝑥 → 𝐴 < 𝑥) ↔ (0 < 𝐴 → 𝐴 < 𝐴))) |
7 | 6 | rspcv 3618 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥) → (0 < 𝐴 → 𝐴 < 𝐴))) |
8 | ltnr 10735 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
9 | 8 | pm2.21d 121 | . . . . . . . 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 2829 | . . . . 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 855 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 < 𝐴 ∨ 0 = 𝐴) → (∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥) → 𝐴 = 0))) |
18 | 3, 17 | sylbid 242 | . 2 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 → (∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥) → 𝐴 = 0))) |
19 | 18 | 3imp 1107 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 class class class wbr 5066 ℝcr 10536 0cc0 10537 < clt 10675 ≤ cle 10676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-addrcl 10598 ax-rnegex 10608 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 |
This theorem is referenced by: (None) |
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