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Mirrors > Home > MPE Home > Th. List > nnaordi | Structured version Visualization version Unicode version |
Description: Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers. (Contributed by NM, 3-Feb-1996.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
nnaordi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn 7075 | . . . . . 6 | |
2 | 1 | ancoms 469 | . . . . 5 |
3 | 2 | adantll 750 | . . . 4 |
4 | nnord 7073 | . . . . . . . . 9 | |
5 | ordsucss 7018 | . . . . . . . . 9 | |
6 | 4, 5 | syl 17 | . . . . . . . 8 |
7 | 6 | ad2antlr 763 | . . . . . . 7 |
8 | peano2b 7081 | . . . . . . . . . 10 | |
9 | oveq2 6658 | . . . . . . . . . . . . . 14 | |
10 | 9 | sseq2d 3633 | . . . . . . . . . . . . 13 |
11 | 10 | imbi2d 330 | . . . . . . . . . . . 12 |
12 | oveq2 6658 | . . . . . . . . . . . . . 14 | |
13 | 12 | sseq2d 3633 | . . . . . . . . . . . . 13 |
14 | 13 | imbi2d 330 | . . . . . . . . . . . 12 |
15 | oveq2 6658 | . . . . . . . . . . . . . 14 | |
16 | 15 | sseq2d 3633 | . . . . . . . . . . . . 13 |
17 | 16 | imbi2d 330 | . . . . . . . . . . . 12 |
18 | oveq2 6658 | . . . . . . . . . . . . . 14 | |
19 | 18 | sseq2d 3633 | . . . . . . . . . . . . 13 |
20 | 19 | imbi2d 330 | . . . . . . . . . . . 12 |
21 | ssid 3624 | . . . . . . . . . . . . 13 | |
22 | 21 | 2a1i 12 | . . . . . . . . . . . 12 |
23 | sssucid 5802 | . . . . . . . . . . . . . . . . 17 | |
24 | sstr2 3610 | . . . . . . . . . . . . . . . . 17 | |
25 | 23, 24 | mpi 20 | . . . . . . . . . . . . . . . 16 |
26 | nnasuc 7686 | . . . . . . . . . . . . . . . . . 18 | |
27 | 26 | ancoms 469 | . . . . . . . . . . . . . . . . 17 |
28 | 27 | sseq2d 3633 | . . . . . . . . . . . . . . . 16 |
29 | 25, 28 | syl5ibr 236 | . . . . . . . . . . . . . . 15 |
30 | 29 | ex 450 | . . . . . . . . . . . . . 14 |
31 | 30 | ad2antrr 762 | . . . . . . . . . . . . 13 |
32 | 31 | a2d 29 | . . . . . . . . . . . 12 |
33 | 11, 14, 17, 20, 22, 32 | findsg 7093 | . . . . . . . . . . 11 |
34 | 33 | exp31 630 | . . . . . . . . . 10 |
35 | 8, 34 | syl5bi 232 | . . . . . . . . 9 |
36 | 35 | com4r 94 | . . . . . . . 8 |
37 | 36 | imp31 448 | . . . . . . 7 |
38 | nnasuc 7686 | . . . . . . . . . 10 | |
39 | 38 | sseq1d 3632 | . . . . . . . . 9 |
40 | ovex 6678 | . . . . . . . . . 10 | |
41 | sucssel 5819 | . . . . . . . . . 10 | |
42 | 40, 41 | ax-mp 5 | . . . . . . . . 9 |
43 | 39, 42 | syl6bi 243 | . . . . . . . 8 |
44 | 43 | adantlr 751 | . . . . . . 7 |
45 | 7, 37, 44 | 3syld 60 | . . . . . 6 |
46 | 45 | imp 445 | . . . . 5 |
47 | 46 | an32s 846 | . . . 4 |
48 | 3, 47 | mpdan 702 | . . 3 |
49 | 48 | ex 450 | . 2 |
50 | 49 | ancoms 469 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 wss 3574 word 5722 csuc 5725 (class class class)co 6650 com 7065 coa 7557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 |
This theorem is referenced by: nnaord 7699 nnmordi 7711 addclpi 9714 addnidpi 9723 |
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