Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fz0 | Structured version Visualization version GIF version |
Description: A finite set of sequential integers is empty if its bounds are not integers. (Contributed by AV, 13-Oct-2018.) |
Ref | Expression |
---|---|
fz0 | ⊢ ((𝑀 ∉ ℤ ∨ 𝑁 ∉ ℤ) → (𝑀...𝑁) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3124 | . . 3 ⊢ (𝑀 ∉ ℤ ↔ ¬ 𝑀 ∈ ℤ) | |
2 | df-nel 3124 | . . 3 ⊢ (𝑁 ∉ ℤ ↔ ¬ 𝑁 ∈ ℤ) | |
3 | 1, 2 | orbi12i 911 | . 2 ⊢ ((𝑀 ∉ ℤ ∨ 𝑁 ∉ ℤ) ↔ (¬ 𝑀 ∈ ℤ ∨ ¬ 𝑁 ∈ ℤ)) |
4 | ianor 978 | . . 3 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ (¬ 𝑀 ∈ ℤ ∨ ¬ 𝑁 ∈ ℤ)) | |
5 | fzf 12886 | . . . . 5 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | |
6 | 5 | fdmi 6510 | . . . 4 ⊢ dom ... = (ℤ × ℤ) |
7 | 6 | ndmov 7318 | . . 3 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ∅) |
8 | 4, 7 | sylbir 237 | . 2 ⊢ ((¬ 𝑀 ∈ ℤ ∨ ¬ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ∅) |
9 | 3, 8 | sylbi 219 | 1 ⊢ ((𝑀 ∉ ℤ ∨ 𝑁 ∉ ℤ) → (𝑀...𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 ∅c0 4279 𝒫 cpw 4525 × cxp 5539 (class class class)co 7142 ℤcz 11968 ...cfz 12882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-1st 7675 df-2nd 7676 df-neg 10859 df-z 11969 df-fz 12883 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |