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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 43546 | . . . . . 6 ⊢ dom ℤ≥ = ℤ | |
2 | 1 | eqcomi 2742 | . . . . 5 ⊢ ℤ = dom ℤ≥ |
3 | 2 | eleq2i 2826 | . . . 4 ⊢ (𝑀 ∈ ℤ ↔ 𝑀 ∈ dom ℤ≥) |
4 | 3 | notbii 320 | . . 3 ⊢ (¬ 𝑀 ∈ ℤ ↔ ¬ 𝑀 ∈ dom ℤ≥) |
5 | 4 | biimpi 215 | . 2 ⊢ (¬ 𝑀 ∈ ℤ → ¬ 𝑀 ∈ dom ℤ≥) |
6 | ndmfv 6878 | . 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 1542 ∈ wcel 2107 ∅c0 4283 dom cdm 5634 ‘cfv 6497 ℤcz 12504 ℤ≥cuz 12768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-cnex 11112 ax-resscn 11113 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-neg 11393 df-z 12505 df-uz 12769 |
This theorem is referenced by: uzn0bi 43780 limsupubuz 44040 climlimsupcex 44096 |
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