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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 3127 | . . 3 ⊢ (𝑀 ∉ ℤ ↔ ¬ 𝑀 ∈ ℤ) | |
2 | df-nel 3127 | . . 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 12899 | . . . . 5 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | |
6 | 5 | fdmi 6527 | . . . 4 ⊢ dom ... = (ℤ × ℤ) |
7 | 6 | ndmov 7335 | . . 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 1536 ∈ wcel 2113 ∉ wnel 3126 ∅c0 4294 𝒫 cpw 4542 × cxp 5556 (class class class)co 7159 ℤcz 11984 ...cfz 12895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-neg 10876 df-z 11985 df-fz 12896 |
This theorem is referenced by: ffz0iswrdOLD 13895 |
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