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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 10873 | . 2 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | nnord 7865 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
3 | ord0eln0 6418 | . . . 4 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
5 | 4 | pm5.32i 573 | . 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 394 ∈ wcel 2104 ≠ wne 2938 ∅c0 4321 Ord word 6362 ωcom 7857 Ncnpi 10841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 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 6366 df-on 6367 df-om 7858 df-ni 10869 |
This theorem is referenced by: addclpi 10889 mulclpi 10890 mulcanpi 10897 addnidpi 10898 ltexpi 10899 ltmpi 10901 indpi 10904 |
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