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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 707 | . . 3 | |
2 | 1cnd 7936 | . . 3 | |
3 | 1, 2 | 2thd 174 | . 2 |
4 | eqeq1 2177 | . . 3 | |
5 | oveq1 5860 | . . . 4 | |
6 | 5 | eleq1d 2239 | . . 3 |
7 | 4, 6 | orbi12d 788 | . 2 |
8 | eqeq1 2177 | . . 3 | |
9 | oveq1 5860 | . . . 4 | |
10 | 9 | eleq1d 2239 | . . 3 |
11 | 8, 10 | orbi12d 788 | . 2 |
12 | eqeq1 2177 | . . 3 | |
13 | oveq1 5860 | . . . 4 | |
14 | 13 | eleq1d 2239 | . . 3 |
15 | 12, 14 | orbi12d 788 | . 2 |
16 | ax-1cn 7867 | . 2 | |
17 | nncn 8886 | . . . . . 6 | |
18 | pncan 8125 | . . . . . 6 | |
19 | 17, 16, 18 | sylancl 411 | . . . . 5 |
20 | id 19 | . . . . 5 | |
21 | 19, 20 | eqeltrd 2247 | . . . 4 |
22 | 21 | olcd 729 | . . 3 |
23 | 22 | a1d 22 | . 2 |
24 | 3, 7, 11, 15, 16, 23 | nnind 8894 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 703 wceq 1348 wcel 2141 (class class class)co 5853 cc 7772 c1 7775 caddc 7777 cmin 8090 cn 8878 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-sub 8092 df-inn 8879 |
This theorem is referenced by: nn1suc 8897 nnsub 8917 nnm1nn0 9176 nn0ge2m1nn 9195 |
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