peano2nn0 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > peano2nn0		Unicode version

Theorem peano2nn0 9113

Description: Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)

**Assertion**
Ref	Expression
peano2nn0

**Proof of Theorem peano2nn0**
Step	Hyp	Ref	Expression
1		1nn0 9089	. 2
2		nn0addcl 9108	. 2 $/\$
3	1, 2	mpan2 422	1

Colors of variables: wff set class

Syntax hints:

wi 4

wcel 2128 (class class class)co 5818

c1 7716

caddc 7718

cn0 9073

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-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-sep 4082 ax-cnex 7806 ax-resscn 7807 ax-1cn 7808 ax-1re 7809 ax-icn 7810 ax-addcl 7811 ax-addrcl 7812 ax-mulcl 7813 ax-addcom 7815 ax-addass 7817 ax-i2m1 7820 ax-0id 7823

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-iota 5132 df-fv 5175 df-ov 5821 df-inn 8817 df-n0 9074

This theorem is referenced by: peano2z 9186 nn0split 10017 fzonn0p1p1 10094 elfzom1p1elfzo 10095 frecfzennn 10307 leexp2r 10455 facdiv 10594 facwordi 10596 faclbnd 10597 faclbnd2 10598 faclbnd3 10599 faclbnd6 10600 bcnp1n 10615 bcp1m1 10621 bcpasc 10622 hashfz 10677 bcxmas 11368 geolim 11390 geo2sum 11393 mertenslemub 11413 mertenslemi1 11414 mertenslem2 11415 mertensabs 11416 efcllemp 11537 eftlub 11569 efsep 11570 effsumlt 11571 nn0ob 11780 nn0oddm1d2 11781 nn0seqcvgd 11898 algcvg 11905 pw2dvdseulemle 12021 2sqpwodd 12030 nonsq 12061 ennnfonelemp1 12107 ennnfonelemkh 12113 ennnfonelemim 12125