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Mirrors > Home > ILE Home > Th. List > uzn0 | GIF version |
Description: The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.) |
Ref | Expression |
---|---|
uzn0 | ⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzf 9329 | . . 3 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
2 | ffn 5272 | . . 3 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ) | |
3 | fvelrnb 5469 | . . 3 ⊢ (ℤ≥ Fn ℤ → (𝑀 ∈ ran ℤ≥ ↔ ∃𝑘 ∈ ℤ (ℤ≥‘𝑘) = 𝑀)) | |
4 | 1, 2, 3 | mp2b 8 | . 2 ⊢ (𝑀 ∈ ran ℤ≥ ↔ ∃𝑘 ∈ ℤ (ℤ≥‘𝑘) = 𝑀) |
5 | uzid 9340 | . . . . 5 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ (ℤ≥‘𝑘)) | |
6 | ne0i 3369 | . . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑘) → (ℤ≥‘𝑘) ≠ ∅) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ (𝑘 ∈ ℤ → (ℤ≥‘𝑘) ≠ ∅) |
8 | neeq1 2321 | . . . 4 ⊢ ((ℤ≥‘𝑘) = 𝑀 → ((ℤ≥‘𝑘) ≠ ∅ ↔ 𝑀 ≠ ∅)) | |
9 | 7, 8 | syl5ibcom 154 | . . 3 ⊢ (𝑘 ∈ ℤ → ((ℤ≥‘𝑘) = 𝑀 → 𝑀 ≠ ∅)) |
10 | 9 | rexlimiv 2543 | . 2 ⊢ (∃𝑘 ∈ ℤ (ℤ≥‘𝑘) = 𝑀 → 𝑀 ≠ ∅) |
11 | 4, 10 | sylbi 120 | 1 ⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 ∃wrex 2417 ∅c0 3363 𝒫 cpw 3510 ran crn 4540 Fn wfn 5118 ⟶wf 5119 ‘cfv 5123 ℤcz 9054 ℤ≥cuz 9326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-neg 7936 df-z 9055 df-uz 9327 |
This theorem is referenced by: (None) |
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