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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 10914 | . 2 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | nnord 7895 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
3 | ord0eln0 6441 | . . . 4 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
5 | 4 | pm5.32i 574 | . 2 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) |
6 | 1, 5 | bitr4i 278 | 1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 Ord word 6385 ωcom 7887 Ncnpi 10882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-om 7888 df-ni 10910 |
This theorem is referenced by: addclpi 10930 mulclpi 10931 mulcanpi 10938 addnidpi 10939 ltexpi 10940 ltmpi 10942 indpi 10945 |
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