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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 2783 | . . 3 ⊢ (𝑀 ∉ ℤ ↔ ¬ 𝑀 ∈ ℤ) | |
2 | df-nel 2783 | . . 3 ⊢ (𝑁 ∉ ℤ ↔ ¬ 𝑁 ∈ ℤ) | |
3 | 1, 2 | orbi12i 542 | . 2 ⊢ ((𝑀 ∉ ℤ ∨ 𝑁 ∉ ℤ) ↔ (¬ 𝑀 ∈ ℤ ∨ ¬ 𝑁 ∈ ℤ)) |
4 | ianor 508 | . . 3 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ (¬ 𝑀 ∈ ℤ ∨ ¬ 𝑁 ∈ ℤ)) | |
5 | fzf 12201 | . . . . 5 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | |
6 | 5 | fdmi 5965 | . . . 4 ⊢ dom ... = (ℤ × ℤ) |
7 | 6 | ndmov 6716 | . . 3 ⊢ (¬ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ∅) |
8 | 4, 7 | sylbir 224 | . 2 ⊢ ((¬ 𝑀 ∈ ℤ ∨ ¬ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ∅) |
9 | 3, 8 | sylbi 206 | 1 ⊢ ((𝑀 ∉ ℤ ∨ 𝑁 ∉ ℤ) → (𝑀...𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 ∅c0 3874 𝒫 cpw 4108 × cxp 5036 (class class class)co 6549 ℤcz 11254 ...cfz 12197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-neg 10148 df-z 11255 df-fz 12198 |
This theorem is referenced by: ffz0iswrd 13187 |
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