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Mirrors > Home > ILE Home > Th. List > nn1m1nn | Unicode version |
Description: Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nn1m1nn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 702 | . . 3 | |
2 | 1cnd 7894 | . . 3 | |
3 | 1, 2 | 2thd 174 | . 2 |
4 | eqeq1 2164 | . . 3 | |
5 | oveq1 5831 | . . . 4 | |
6 | 5 | eleq1d 2226 | . . 3 |
7 | 4, 6 | orbi12d 783 | . 2 |
8 | eqeq1 2164 | . . 3 | |
9 | oveq1 5831 | . . . 4 | |
10 | 9 | eleq1d 2226 | . . 3 |
11 | 8, 10 | orbi12d 783 | . 2 |
12 | eqeq1 2164 | . . 3 | |
13 | oveq1 5831 | . . . 4 | |
14 | 13 | eleq1d 2226 | . . 3 |
15 | 12, 14 | orbi12d 783 | . 2 |
16 | ax-1cn 7825 | . 2 | |
17 | nncn 8841 | . . . . . 6 | |
18 | pncan 8081 | . . . . . 6 | |
19 | 17, 16, 18 | sylancl 410 | . . . . 5 |
20 | id 19 | . . . . 5 | |
21 | 19, 20 | eqeltrd 2234 | . . . 4 |
22 | 21 | olcd 724 | . . 3 |
23 | 22 | a1d 22 | . 2 |
24 | 3, 7, 11, 15, 16, 23 | nnind 8849 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 698 wceq 1335 wcel 2128 (class class class)co 5824 cc 7730 c1 7733 caddc 7735 cmin 8046 cn 8833 |
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 604 ax-in2 605 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-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-sub 8048 df-inn 8834 |
This theorem is referenced by: nn1suc 8852 nnsub 8872 nnm1nn0 9131 nn0ge2m1nn 9150 |
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