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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 40877 | . . . . . 6 ⊢ dom ℤ≥ = ℤ | |
2 | 1 | eqcomi 2781 | . . . . 5 ⊢ ℤ = dom ℤ≥ |
3 | 2 | eleq2i 2851 | . . . 4 ⊢ (𝑀 ∈ ℤ ↔ 𝑀 ∈ dom ℤ≥) |
4 | 3 | notbii 312 | . . 3 ⊢ (¬ 𝑀 ∈ ℤ ↔ ¬ 𝑀 ∈ dom ℤ≥) |
5 | 4 | biimpi 208 | . 2 ⊢ (¬ 𝑀 ∈ ℤ → ¬ 𝑀 ∈ dom ℤ≥) |
6 | ndmfv 6523 | . 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 1507 ∈ wcel 2048 ∅c0 4173 dom cdm 5400 ‘cfv 6182 ℤcz 11786 ℤ≥cuz 12051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-cnex 10383 ax-resscn 10384 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-fv 6190 df-ov 6973 df-neg 10665 df-z 11787 df-uz 12052 |
This theorem is referenced by: uzn0bi 41112 limsupubuz 41371 climlimsupcex 41427 |
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