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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 12879 | . . 3 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
2 | ffn 6737 | . . 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 12891 | . . . . 5 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ (ℤ≥‘𝑘)) | |
6 | 5 | ne0d 4348 | . . . 4 ⊢ (𝑘 ∈ ℤ → (ℤ≥‘𝑘) ≠ ∅) |
7 | neeq1 3001 | . . . 4 ⊢ ((ℤ≥‘𝑘) = 𝑀 → ((ℤ≥‘𝑘) ≠ ∅ ↔ 𝑀 ≠ ∅)) | |
8 | 6, 7 | syl5ibcom 245 | . . 3 ⊢ (𝑘 ∈ ℤ → ((ℤ≥‘𝑘) = 𝑀 → 𝑀 ≠ ∅)) |
9 | 8 | rexlimiv 3146 | . 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 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 ∅c0 4339 𝒫 cpw 4605 ran crn 5690 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 ℤcz 12611 ℤ≥cuz 12876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-neg 11493 df-z 12612 df-uz 12877 |
This theorem is referenced by: heibor1lem 37796 |
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