![]() |
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 3092 | . . 3 ⊢ (𝑀 ∉ ℤ ↔ ¬ 𝑀 ∈ ℤ) | |
2 | df-nel 3092 | . . 3 ⊢ (𝑁 ∉ ℤ ↔ ¬ 𝑁 ∈ ℤ) | |
3 | 1, 2 | orbi12i 912 | . 2 ⊢ ((𝑀 ∉ ℤ ∨ 𝑁 ∉ ℤ) ↔ (¬ 𝑀 ∈ ℤ ∨ ¬ 𝑁 ∈ ℤ)) |
4 | ianor 979 | . . 3 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ (¬ 𝑀 ∈ ℤ ∨ ¬ 𝑁 ∈ ℤ)) | |
5 | fzf 12889 | . . . . 5 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | |
6 | 5 | fdmi 6498 | . . . 4 ⊢ dom ... = (ℤ × ℤ) |
7 | 6 | ndmov 7312 | . . 3 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ∅) |
8 | 4, 7 | sylbir 238 | . 2 ⊢ ((¬ 𝑀 ∈ ℤ ∨ ¬ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ∅) |
9 | 3, 8 | sylbi 220 | 1 ⊢ ((𝑀 ∉ ℤ ∨ 𝑁 ∉ ℤ) → (𝑀...𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 ∅c0 4243 𝒫 cpw 4497 × cxp 5517 (class class class)co 7135 ℤcz 11969 ...cfz 12885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-neg 10862 df-z 11970 df-fz 12886 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |