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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 44580 | . . . . . 6 ⊢ dom ℤ≥ = ℤ | |
2 | 1 | eqcomi 2736 | . . . . 5 ⊢ ℤ = dom ℤ≥ |
3 | 2 | eleq2i 2820 | . . . 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 6926 | . 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 1534 ∈ wcel 2099 ∅c0 4318 dom cdm 5672 ‘cfv 6542 ℤcz 12582 ℤ≥cuz 12846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-cnex 11188 ax-resscn 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-neg 11471 df-z 12583 df-uz 12847 |
This theorem is referenced by: uzn0bi 44813 limsupubuz 45073 climlimsupcex 45129 |
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