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Mirrors > Home > MPE Home > Th. List > elni2 | Structured version Visualization version GIF version |
Description: Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elni2 | ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elni 10867 | . 2 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | nnord 7859 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
3 | ord0eln0 6416 | . . . 4 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
5 | 4 | pm5.32i 575 | . 2 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) |
6 | 1, 5 | bitr4i 277 | 1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ≠ wne 2940 ∅c0 4321 Ord word 6360 ωcom 7851 Ncnpi 10835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-ord 6364 df-on 6365 df-om 7852 df-ni 10863 |
This theorem is referenced by: addclpi 10883 mulclpi 10884 mulcanpi 10891 addnidpi 10892 ltexpi 10893 ltmpi 10895 indpi 10898 |
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