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Mirrors > Home > ILE Home > Th. List > peano2nn0 | GIF version |
Description: Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
peano2nn0 | ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 8897 | . 2 ⊢ 1 ∈ ℕ0 | |
2 | nn0addcl 8916 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ0) | |
3 | 1, 2 | mpan2 419 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1463 (class class class)co 5728 1c1 7548 + caddc 7550 ℕ0cn0 8881 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-addcom 7645 ax-addass 7647 ax-i2m1 7650 ax-0id 7653 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-iota 5046 df-fv 5089 df-ov 5731 df-inn 8631 df-n0 8882 |
This theorem is referenced by: peano2z 8994 nn0split 9806 fzonn0p1p1 9883 elfzom1p1elfzo 9884 frecfzennn 10092 leexp2r 10240 facdiv 10377 facwordi 10379 faclbnd 10380 faclbnd2 10381 faclbnd3 10382 faclbnd6 10383 bcnp1n 10398 bcp1m1 10404 bcpasc 10405 hashfz 10460 bcxmas 11150 geolim 11172 geo2sum 11175 mertenslemub 11195 mertenslemi1 11196 mertenslem2 11197 mertensabs 11198 efcllemp 11215 eftlub 11247 efsep 11248 effsumlt 11249 nn0ob 11453 nn0oddm1d2 11454 nn0seqcvgd 11568 algcvg 11575 pw2dvdseulemle 11690 2sqpwodd 11699 nonsq 11730 ennnfonelemp1 11764 ennnfonelemkh 11770 ennnfonelemim 11782 |
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