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Mirrors > Home > ILE Home > Th. List > pinn | GIF version |
Description: A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) |
Ref | Expression |
---|---|
pinn | ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ni 7105 | . . 3 ⊢ N = (ω ∖ {∅}) | |
2 | difss 3197 | . . 3 ⊢ (ω ∖ {∅}) ⊆ ω | |
3 | 1, 2 | eqsstri 3124 | . 2 ⊢ N ⊆ ω |
4 | 3 | sseli 3088 | 1 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ∖ cdif 3063 ∅c0 3358 {csn 3522 ωcom 4499 Ncnpi 7073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-in 3072 df-ss 3079 df-ni 7105 |
This theorem is referenced by: pion 7111 piord 7112 elni2 7115 mulidpi 7119 ltsopi 7121 pitric 7122 pitri3or 7123 ltdcpi 7124 addclpi 7128 mulclpi 7129 addcompig 7130 addasspig 7131 mulcompig 7132 mulasspig 7133 distrpig 7134 addcanpig 7135 mulcanpig 7136 addnidpig 7137 ltexpi 7138 ltapig 7139 ltmpig 7140 nnppipi 7144 enqdc 7162 archnqq 7218 prarloclemarch2 7220 enq0enq 7232 enq0sym 7233 enq0ref 7234 enq0tr 7235 nqnq0pi 7239 nqnq0 7242 addcmpblnq0 7244 mulcmpblnq0 7245 mulcanenq0ec 7246 addclnq0 7252 nqpnq0nq 7254 nqnq0a 7255 nqnq0m 7256 nq0m0r 7257 nq0a0 7258 nnanq0 7259 distrnq0 7260 mulcomnq0 7261 addassnq0lemcl 7262 addassnq0 7263 nq02m 7266 prarloclemlt 7294 prarloclemn 7300 |
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