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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uzn0bi | Structured version Visualization version GIF version |
Description: The upper integers function needs to be applied to an integer, in order to return a nonempty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
uzn0bi | ⊢ ((ℤ≥‘𝑀) ≠ ∅ ↔ 𝑀 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uz0 42049 | . . . 4 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) | |
2 | 1 | adantl 485 | . . 3 ⊢ (((ℤ≥‘𝑀) ≠ ∅ ∧ ¬ 𝑀 ∈ ℤ) → (ℤ≥‘𝑀) = ∅) |
3 | neneq 2993 | . . . 4 ⊢ ((ℤ≥‘𝑀) ≠ ∅ → ¬ (ℤ≥‘𝑀) = ∅) | |
4 | 3 | adantr 484 | . . 3 ⊢ (((ℤ≥‘𝑀) ≠ ∅ ∧ ¬ 𝑀 ∈ ℤ) → ¬ (ℤ≥‘𝑀) = ∅) |
5 | 2, 4 | condan 817 | . 2 ⊢ ((ℤ≥‘𝑀) ≠ ∅ → 𝑀 ∈ ℤ) |
6 | id 22 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
7 | eqid 2798 | . . 3 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
8 | 6, 7 | uzn0d 42062 | . 2 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ≠ ∅) |
9 | 5, 8 | impbii 212 | 1 ⊢ ((ℤ≥‘𝑀) ≠ ∅ ↔ 𝑀 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 ‘cfv 6324 ℤcz 11969 ℤ≥cuz 12231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-z 11970 df-uz 12232 |
This theorem is referenced by: (None) |
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