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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 44675 | . . . 4 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) | |
2 | 1 | adantl 481 | . . 3 ⊢ (((ℤ≥‘𝑀) ≠ ∅ ∧ ¬ 𝑀 ∈ ℤ) → (ℤ≥‘𝑀) = ∅) |
3 | neneq 2940 | . . . 4 ⊢ ((ℤ≥‘𝑀) ≠ ∅ → ¬ (ℤ≥‘𝑀) = ∅) | |
4 | 3 | adantr 480 | . . 3 ⊢ (((ℤ≥‘𝑀) ≠ ∅ ∧ ¬ 𝑀 ∈ ℤ) → ¬ (ℤ≥‘𝑀) = ∅) |
5 | 2, 4 | condan 815 | . 2 ⊢ ((ℤ≥‘𝑀) ≠ ∅ → 𝑀 ∈ ℤ) |
6 | id 22 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℤ) | |
7 | eqid 2726 | . . 3 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
8 | 6, 7 | uzn0d 44688 | . 2 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ≠ ∅) |
9 | 5, 8 | impbii 208 | 1 ⊢ ((ℤ≥‘𝑀) ≠ ∅ ↔ 𝑀 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∅c0 4317 ‘cfv 6536 ℤcz 12559 ℤ≥cuz 12823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-neg 11448 df-z 12560 df-uz 12824 |
This theorem is referenced by: (None) |
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