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Mirrors > Home > ILE Home > Th. List > elni | GIF version |
Description: Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) |
Ref | Expression |
---|---|
elni | ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ni 7136 | . . 3 ⊢ N = (ω ∖ {∅}) | |
2 | 1 | eleq2i 2207 | . 2 ⊢ (𝐴 ∈ N ↔ 𝐴 ∈ (ω ∖ {∅})) |
3 | eldifsn 3658 | . 2 ⊢ (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
4 | 2, 3 | bitri 183 | 1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1481 ≠ wne 2309 ∖ cdif 3073 ∅c0 3368 {csn 3532 ωcom 4512 Ncnpi 7104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-v 2691 df-dif 3078 df-sn 3538 df-ni 7136 |
This theorem is referenced by: 0npi 7145 elni2 7146 1pi 7147 addclpi 7159 mulclpi 7160 nlt1pig 7173 indpi 7174 nqnq0pi 7270 prarloclemcalc 7334 |
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