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| Mirrors > Home > MPE Home > Th. List > uzn0 | Structured version Visualization version 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 12881 | . . 3 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
| 2 | ffn 6736 | . . 3 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ) | |
| 3 | fvelrnb 6969 | . . 3 ⊢ (ℤ≥ Fn ℤ → (𝑀 ∈ ran ℤ≥ ↔ ∃𝑘 ∈ ℤ (ℤ≥‘𝑘) = 𝑀)) | |
| 4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ (𝑀 ∈ ran ℤ≥ ↔ ∃𝑘 ∈ ℤ (ℤ≥‘𝑘) = 𝑀) |
| 5 | uzid 12893 | . . . . 5 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ (ℤ≥‘𝑘)) | |
| 6 | 5 | ne0d 4342 | . . . 4 ⊢ (𝑘 ∈ ℤ → (ℤ≥‘𝑘) ≠ ∅) |
| 7 | neeq1 3003 | . . . 4 ⊢ ((ℤ≥‘𝑘) = 𝑀 → ((ℤ≥‘𝑘) ≠ ∅ ↔ 𝑀 ≠ ∅)) | |
| 8 | 6, 7 | syl5ibcom 245 | . . 3 ⊢ (𝑘 ∈ ℤ → ((ℤ≥‘𝑘) = 𝑀 → 𝑀 ≠ ∅)) |
| 9 | 8 | rexlimiv 3148 | . 2 ⊢ (∃𝑘 ∈ ℤ (ℤ≥‘𝑘) = 𝑀 → 𝑀 ≠ ∅) |
| 10 | 4, 9 | sylbi 217 | 1 ⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ∅c0 4333 𝒫 cpw 4600 ran crn 5686 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 ℤcz 12613 ℤ≥cuz 12878 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 |
| This theorem is referenced by: heibor1lem 37816 |
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