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Mirrors > Home > MPE Home > Th. List > Mathboxes > elzdif0 | Structured version Visualization version GIF version |
Description: Lemma for qqhval2 33491. (Contributed by Thierry Arnoux, 29-Oct-2017.) |
Ref | Expression |
---|---|
elzdif0 | ⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsnneq 4789 | . 2 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → ¬ 𝑀 = 0) | |
2 | eldifi 4121 | . . . . 5 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → 𝑀 ∈ ℤ) | |
3 | elz 12561 | . . . . 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 495 | . . 3 ⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 = 0 ∨ 𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ)) |
6 | 3orass 1087 | . . 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 885 | . 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 844 ∨ w3o 1083 = wceq 1533 ∈ wcel 2098 ∖ cdif 3940 {csn 4623 ℝcr 11108 0cc0 11109 -cneg 11446 ℕcn 12213 ℤcz 12559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6488 df-fv 6544 df-ov 7407 df-neg 11448 df-z 12560 |
This theorem is referenced by: (None) |
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