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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 10628 | . . 3 ⊢ N = (ω ∖ {∅}) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ N ↔ 𝐴 ∈ (ω ∖ {∅})) |
3 | eldifsn 4720 | . 2 ⊢ (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
4 | 2, 3 | bitri 274 | 1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 ∅c0 4256 {csn 4561 ωcom 7712 Ncnpi 10600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-v 3434 df-dif 3890 df-sn 4562 df-ni 10628 |
This theorem is referenced by: elni2 10633 0npi 10638 1pi 10639 addclpi 10648 mulclpi 10649 nlt1pi 10662 indpi 10663 |
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