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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 9981 | . . 3 ⊢ N = (ω ∖ {∅}) | |
2 | 1 | eleq2i 2869 | . 2 ⊢ (𝐴 ∈ N ↔ 𝐴 ∈ (ω ∖ {∅})) |
3 | eldifsn 4505 | . 2 ⊢ (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
4 | 2, 3 | bitri 267 | 1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 385 ∈ wcel 2157 ≠ wne 2970 ∖ cdif 3765 ∅c0 4114 {csn 4367 ωcom 7298 Ncnpi 9953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2776 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-v 3386 df-dif 3771 df-sn 4368 df-ni 9981 |
This theorem is referenced by: elni2 9986 0npi 9991 1pi 9992 addclpi 10001 mulclpi 10002 nlt1pi 10015 indpi 10016 |
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