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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 10376 | . 2 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | nnord 7607 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
3 | ord0eln0 6226 | . . . 4 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
5 | 4 | pm5.32i 578 | . 2 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) |
6 | 1, 5 | bitr4i 281 | 1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2114 ≠ wne 2934 ∅c0 4211 Ord word 6171 ωcom 7599 Ncnpi 10344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-11 2162 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-ord 6175 df-on 6176 df-om 7600 df-ni 10372 |
This theorem is referenced by: addclpi 10392 mulclpi 10393 mulcanpi 10400 addnidpi 10401 ltexpi 10402 ltmpi 10404 indpi 10407 |
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