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| Mirrors > Home > MPE Home > Th. List > elni | Structured version Visualization version GIF version | ||
| Description: Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| elni | ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ni 10912 | . . 3 ⊢ N = (ω ∖ {∅}) | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ N ↔ 𝐴 ∈ (ω ∖ {∅})) |
| 3 | eldifsn 4786 | . 2 ⊢ (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
| 4 | 2, 3 | bitri 275 | 1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ∅c0 4333 {csn 4626 ωcom 7887 Ncnpi 10884 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-sn 4627 df-ni 10912 |
| This theorem is referenced by: elni2 10917 0npi 10922 1pi 10923 addclpi 10932 mulclpi 10933 nlt1pi 10946 indpi 10947 |
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