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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 10016 | . . 3 ⊢ N = (ω ∖ {∅}) | |
2 | 1 | eleq2i 2898 | . 2 ⊢ (𝐴 ∈ N ↔ 𝐴 ∈ (ω ∖ {∅})) |
3 | eldifsn 4538 | . 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 386 ∈ wcel 2164 ≠ wne 2999 ∖ cdif 3795 ∅c0 4146 {csn 4399 ωcom 7331 Ncnpi 9988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-v 3416 df-dif 3801 df-sn 4400 df-ni 10016 |
This theorem is referenced by: elni2 10021 0npi 10026 1pi 10027 addclpi 10036 mulclpi 10037 nlt1pi 10050 indpi 10051 |
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