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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 9460 | . . 3 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
2 | ffn 5331 | . . 3 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ) | |
3 | fvelrnb 5528 | . . 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 9471 | . . . . 5 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ (ℤ≥‘𝑘)) | |
6 | ne0i 3410 | . . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑘) → (ℤ≥‘𝑘) ≠ ∅) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ (𝑘 ∈ ℤ → (ℤ≥‘𝑘) ≠ ∅) |
8 | neeq1 2347 | . . . 4 ⊢ ((ℤ≥‘𝑘) = 𝑀 → ((ℤ≥‘𝑘) ≠ ∅ ↔ 𝑀 ≠ ∅)) | |
9 | 7, 8 | syl5ibcom 154 | . . 3 ⊢ (𝑘 ∈ ℤ → ((ℤ≥‘𝑘) = 𝑀 → 𝑀 ≠ ∅)) |
10 | 9 | rexlimiv 2575 | . 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 1342 ∈ wcel 2135 ≠ wne 2334 ∃wrex 2443 ∅c0 3404 𝒫 cpw 3553 ran crn 4599 Fn wfn 5177 ⟶wf 5178 ‘cfv 5182 ℤcz 9182 ℤ≥cuz 9457 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-ltirr 7856 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-neg 8063 df-z 9183 df-uz 9458 |
This theorem is referenced by: (None) |
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