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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 10292 | . 2 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | |
2 | nnord 7582 | . . . 4 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
3 | ord0eln0 6239 | . . . 4 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
5 | 4 | pm5.32i 577 | . 2 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) |
6 | 1, 5 | bitr4i 280 | 1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ≠ wne 3016 ∅c0 4290 Ord word 6184 ωcom 7574 Ncnpi 10260 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-tr 5165 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-om 7575 df-ni 10288 |
This theorem is referenced by: addclpi 10308 mulclpi 10309 mulcanpi 10316 addnidpi 10317 ltexpi 10318 ltmpi 10320 indpi 10323 |
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