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 41706 | . . . 4 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) | |
2 | 1 | adantl 484 | . . 3 ⊢ (((ℤ≥‘𝑀) ≠ ∅ ∧ ¬ 𝑀 ∈ ℤ) → (ℤ≥‘𝑀) = ∅) |
3 | neneq 3022 | . . . 4 ⊢ ((ℤ≥‘𝑀) ≠ ∅ → ¬ (ℤ≥‘𝑀) = ∅) | |
4 | 3 | adantr 483 | . . 3 ⊢ (((ℤ≥‘𝑀) ≠ ∅ ∧ ¬ 𝑀 ∈ ℤ) → ¬ (ℤ≥‘𝑀) = ∅) |
5 | 2, 4 | condan 816 | . 2 ⊢ ((ℤ≥‘𝑀) ≠ ∅ → 𝑀 ∈ ℤ) |
6 | id 22 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
7 | eqid 2821 | . . 3 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
8 | 6, 7 | uzn0d 41719 | . 2 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ≠ ∅) |
9 | 5, 8 | impbii 211 | 1 ⊢ ((ℤ≥‘𝑀) ≠ ∅ ↔ 𝑀 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 ‘cfv 6355 ℤcz 11982 ℤ≥cuz 12244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-neg 10873 df-z 11983 df-uz 12245 |
This theorem is referenced by: (None) |
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