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| Description: Lemma for qqhval2 33984. (Contributed by Thierry Arnoux, 29-Oct-2017.) | 
| Ref | Expression | 
|---|---|
| elzdif0 | ⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eldifsnneq 4790 | . 2 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → ¬ 𝑀 = 0) | |
| 2 | eldifi 4130 | . . . . 5 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → 𝑀 ∈ ℤ) | |
| 3 | elz 12617 | . . . . 5 ⊢ (𝑀 ∈ ℤ ↔ (𝑀 ∈ ℝ ∧ (𝑀 = 0 ∨ 𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))) | |
| 4 | 2, 3 | sylib 218 | . . . 4 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℝ ∧ (𝑀 = 0 ∨ 𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))) | 
| 5 | 4 | simprd 495 | . . 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 218 | . 2 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 = 0 ∨ (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))) | 
| 8 | orel1 888 | . 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 395 ∨ wo 847 ∨ w3o 1085 = wceq 1539 ∈ wcel 2107 ∖ cdif 3947 {csn 4625 ℝcr 11155 0cc0 11156 -cneg 11494 ℕcn 12267 ℤcz 12615 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-neg 11496 df-z 12616 | 
| This theorem is referenced by: (None) | 
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