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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 12240 | . . 3 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
2 | ffn 6508 | . . 3 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ) | |
3 | fvelrnb 6720 | . . 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 12252 | . . . . 5 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ (ℤ≥‘𝑘)) | |
6 | 5 | ne0d 4300 | . . . 4 ⊢ (𝑘 ∈ ℤ → (ℤ≥‘𝑘) ≠ ∅) |
7 | neeq1 3078 | . . . 4 ⊢ ((ℤ≥‘𝑘) = 𝑀 → ((ℤ≥‘𝑘) ≠ ∅ ↔ 𝑀 ≠ ∅)) | |
8 | 6, 7 | syl5ibcom 247 | . . 3 ⊢ (𝑘 ∈ ℤ → ((ℤ≥‘𝑘) = 𝑀 → 𝑀 ≠ ∅)) |
9 | 8 | rexlimiv 3280 | . 2 ⊢ (∃𝑘 ∈ ℤ (ℤ≥‘𝑘) = 𝑀 → 𝑀 ≠ ∅) |
10 | 4, 9 | sylbi 219 | 1 ⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 ∅c0 4290 𝒫 cpw 4538 ran crn 5550 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 ℤcz 11975 ℤ≥cuz 12237 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-neg 10867 df-z 11976 df-uz 12238 |
This theorem is referenced by: heibor1lem 35081 |
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