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Mirrors > Home > ILE Home > Th. List > ap0gt0 | GIF version |
Description: A nonnegative number is apart from zero if and only if it is positive. (Contributed by Jim Kingdon, 11-Aug-2021.) |
Ref | Expression |
---|---|
ap0gt0 | ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 # 0 ↔ 0 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 7971 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ∈ ℝ) | |
2 | reaplt 8558 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 # 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴))) | |
3 | 1, 2 | syldan 282 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 # 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴))) |
4 | simpr 110 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ 𝐴) | |
5 | simpl 109 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ) | |
6 | 1, 5 | lenltd 8088 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0)) |
7 | 4, 6 | mpbid 147 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ¬ 𝐴 < 0) |
8 | biorf 745 | . . 3 ⊢ (¬ 𝐴 < 0 → (0 < 𝐴 ↔ (𝐴 < 0 ∨ 0 < 𝐴))) | |
9 | 7, 8 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0 < 𝐴 ↔ (𝐴 < 0 ∨ 0 < 𝐴))) |
10 | 3, 9 | bitr4d 191 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 # 0 ↔ 0 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 ∈ wcel 2158 class class class wbr 4015 ℝcr 7823 0cc0 7824 < clt 8005 ≤ cle 8006 # cap 8551 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 |
This theorem is referenced by: ap0gt0d 8611 fihashneq0 10787 mul0inf 11262 |
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