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Mirrors > Home > MPE Home > Th. List > Mathboxes > elzdif0 | Structured version Visualization version GIF version |
Description: Lemma for qqhval2 31940. (Contributed by Thierry Arnoux, 29-Oct-2017.) |
Ref | Expression |
---|---|
elzdif0 | ⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsnneq 4724 | . 2 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → ¬ 𝑀 = 0) | |
2 | eldifi 4060 | . . . . 5 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → 𝑀 ∈ ℤ) | |
3 | elz 12331 | . . . . 5 ⊢ (𝑀 ∈ ℤ ↔ (𝑀 ∈ ℝ ∧ (𝑀 = 0 ∨ 𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))) | |
4 | 2, 3 | sylib 217 | . . . 4 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℝ ∧ (𝑀 = 0 ∨ 𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))) |
5 | 4 | simprd 496 | . . 3 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 = 0 ∨ 𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)) |
6 | 3orass 1089 | . . 3 ⊢ ((𝑀 = 0 ∨ 𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ) ↔ (𝑀 = 0 ∨ (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))) | |
7 | 5, 6 | sylib 217 | . 2 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 = 0 ∨ (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))) |
8 | orel1 886 | . 2 ⊢ (¬ 𝑀 = 0 → ((𝑀 = 0 ∨ (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))) | |
9 | 1, 7, 8 | sylc 65 | 1 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 844 ∨ w3o 1085 = wceq 1539 ∈ wcel 2106 ∖ cdif 3883 {csn 4561 ℝcr 10880 0cc0 10881 -cneg 11216 ℕcn 11983 ℤcz 12329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-iota 6384 df-fv 6434 df-ov 7270 df-neg 11218 df-z 12330 |
This theorem is referenced by: (None) |
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