Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uz0 | Structured version Visualization version GIF version |
Description: The upper integers function applied to a non-integer, is the empty set. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
uz0 | ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmuz 41502 | . . . . . 6 ⊢ dom ℤ≥ = ℤ | |
2 | 1 | eqcomi 2830 | . . . . 5 ⊢ ℤ = dom ℤ≥ |
3 | 2 | eleq2i 2904 | . . . 4 ⊢ (𝑀 ∈ ℤ ↔ 𝑀 ∈ dom ℤ≥) |
4 | 3 | notbii 322 | . . 3 ⊢ (¬ 𝑀 ∈ ℤ ↔ ¬ 𝑀 ∈ dom ℤ≥) |
5 | 4 | biimpi 218 | . 2 ⊢ (¬ 𝑀 ∈ ℤ → ¬ 𝑀 ∈ dom ℤ≥) |
6 | ndmfv 6699 | . 2 ⊢ (¬ 𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) = ∅) | |
7 | 5, 6 | syl 17 | 1 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 ∅c0 4290 dom cdm 5554 ‘cfv 6354 ℤcz 11980 ℤ≥cuz 12242 |
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 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-cnex 10592 ax-resscn 10593 |
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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-neg 10872 df-z 11981 df-uz 12243 |
This theorem is referenced by: uzn0bi 41733 limsupubuz 41992 climlimsupcex 42048 |
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