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 42448 | . . . . . 6 ⊢ dom ℤ≥ = ℤ | |
2 | 1 | eqcomi 2746 | . . . . 5 ⊢ ℤ = dom ℤ≥ |
3 | 2 | eleq2i 2829 | . . . 4 ⊢ (𝑀 ∈ ℤ ↔ 𝑀 ∈ dom ℤ≥) |
4 | 3 | notbii 323 | . . 3 ⊢ (¬ 𝑀 ∈ ℤ ↔ ¬ 𝑀 ∈ dom ℤ≥) |
5 | 4 | biimpi 219 | . 2 ⊢ (¬ 𝑀 ∈ ℤ → ¬ 𝑀 ∈ dom ℤ≥) |
6 | ndmfv 6744 | . 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 1543 ∈ wcel 2110 ∅c0 4234 dom cdm 5548 ‘cfv 6377 ℤcz 12173 ℤ≥cuz 12435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pr 5319 ax-cnex 10782 ax-resscn 10783 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-fv 6385 df-ov 7213 df-neg 11062 df-z 12174 df-uz 12436 |
This theorem is referenced by: uzn0bi 42672 limsupubuz 42927 climlimsupcex 42983 |
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