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Mirrors > Home > ILE Home > Th. List > nnsucsssuc | Unicode version |
Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4235, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4252. (Contributed by Jim Kingdon, 25-Aug-2019.) |
Ref | Expression |
---|---|
nnsucsssuc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 2966 | . . . . . 6 | |
2 | suceq 4139 | . . . . . . 7 | |
3 | 2 | sseq1d 2972 | . . . . . 6 |
4 | 1, 3 | imbi12d 223 | . . . . 5 |
5 | 4 | imbi2d 219 | . . . 4 |
6 | sseq1 2966 | . . . . . 6 | |
7 | suceq 4139 | . . . . . . 7 | |
8 | 7 | sseq1d 2972 | . . . . . 6 |
9 | 6, 8 | imbi12d 223 | . . . . 5 |
10 | sseq1 2966 | . . . . . 6 | |
11 | suceq 4139 | . . . . . . 7 | |
12 | 11 | sseq1d 2972 | . . . . . 6 |
13 | 10, 12 | imbi12d 223 | . . . . 5 |
14 | sseq1 2966 | . . . . . 6 | |
15 | suceq 4139 | . . . . . . 7 | |
16 | 15 | sseq1d 2972 | . . . . . 6 |
17 | 14, 16 | imbi12d 223 | . . . . 5 |
18 | peano3 4319 | . . . . . . . . 9 | |
19 | 18 | neneqd 2226 | . . . . . . . 8 |
20 | peano2 4318 | . . . . . . . . . 10 | |
21 | 0elnn 4340 | . . . . . . . . . 10 | |
22 | 20, 21 | syl 14 | . . . . . . . . 9 |
23 | 22 | ord 643 | . . . . . . . 8 |
24 | 19, 23 | mpd 13 | . . . . . . 7 |
25 | nnord 4334 | . . . . . . . 8 | |
26 | ordsucim 4226 | . . . . . . . 8 | |
27 | 0ex 3884 | . . . . . . . . 9 | |
28 | ordelsuc 4231 | . . . . . . . . 9 | |
29 | 27, 28 | mpan 400 | . . . . . . . 8 |
30 | 25, 26, 29 | 3syl 17 | . . . . . . 7 |
31 | 24, 30 | mpbid 135 | . . . . . 6 |
32 | 31 | a1d 22 | . . . . 5 |
33 | simp3 906 | . . . . . . . . . 10 | |
34 | simp1l 928 | . . . . . . . . . . 11 | |
35 | simp1r 929 | . . . . . . . . . . . 12 | |
36 | 35, 25 | syl 14 | . . . . . . . . . . 11 |
37 | ordelsuc 4231 | . . . . . . . . . . 11 | |
38 | 34, 36, 37 | syl2anc 391 | . . . . . . . . . 10 |
39 | 33, 38 | mpbird 156 | . . . . . . . . 9 |
40 | nnsucelsuc 6070 | . . . . . . . . . 10 | |
41 | 35, 40 | syl 14 | . . . . . . . . 9 |
42 | 39, 41 | mpbid 135 | . . . . . . . 8 |
43 | peano2 4318 | . . . . . . . . . 10 | |
44 | 34, 43 | syl 14 | . . . . . . . . 9 |
45 | 36, 26 | syl 14 | . . . . . . . . 9 |
46 | ordelsuc 4231 | . . . . . . . . 9 | |
47 | 44, 45, 46 | syl2anc 391 | . . . . . . . 8 |
48 | 42, 47 | mpbid 135 | . . . . . . 7 |
49 | 48 | 3expia 1106 | . . . . . 6 |
50 | 49 | exp31 346 | . . . . 5 |
51 | 9, 13, 17, 32, 50 | finds2 4324 | . . . 4 |
52 | 5, 51 | vtoclga 2619 | . . 3 |
53 | 52 | imp 115 | . 2 |
54 | nnon 4332 | . . 3 | |
55 | onsucsssucr 4235 | . . 3 | |
56 | 54, 25, 55 | syl2an 273 | . 2 |
57 | 53, 56 | impbid 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 w3a 885 wceq 1243 wcel 1393 cvv 2557 wss 2917 c0 3224 word 4099 con0 4100 csuc 4102 com 4313 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 |
This theorem is referenced by: nnaword 6084 |
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