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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 4433, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4450. (Contributed by Jim Kingdon, 25-Aug-2019.) |
Ref | Expression |
---|---|
nnsucsssuc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3125 | . . . . . 6 | |
2 | suceq 4332 | . . . . . . 7 | |
3 | 2 | sseq1d 3131 | . . . . . 6 |
4 | 1, 3 | imbi12d 233 | . . . . 5 |
5 | 4 | imbi2d 229 | . . . 4 |
6 | sseq1 3125 | . . . . . 6 | |
7 | suceq 4332 | . . . . . . 7 | |
8 | 7 | sseq1d 3131 | . . . . . 6 |
9 | 6, 8 | imbi12d 233 | . . . . 5 |
10 | sseq1 3125 | . . . . . 6 | |
11 | suceq 4332 | . . . . . . 7 | |
12 | 11 | sseq1d 3131 | . . . . . 6 |
13 | 10, 12 | imbi12d 233 | . . . . 5 |
14 | sseq1 3125 | . . . . . 6 | |
15 | suceq 4332 | . . . . . . 7 | |
16 | 15 | sseq1d 3131 | . . . . . 6 |
17 | 14, 16 | imbi12d 233 | . . . . 5 |
18 | peano3 4518 | . . . . . . . . 9 | |
19 | 18 | neneqd 2330 | . . . . . . . 8 |
20 | peano2 4517 | . . . . . . . . . 10 | |
21 | 0elnn 4540 | . . . . . . . . . 10 | |
22 | 20, 21 | syl 14 | . . . . . . . . 9 |
23 | 22 | ord 714 | . . . . . . . 8 |
24 | 19, 23 | mpd 13 | . . . . . . 7 |
25 | nnord 4533 | . . . . . . . 8 | |
26 | ordsucim 4424 | . . . . . . . 8 | |
27 | 0ex 4063 | . . . . . . . . 9 | |
28 | ordelsuc 4429 | . . . . . . . . 9 | |
29 | 27, 28 | mpan 421 | . . . . . . . 8 |
30 | 25, 26, 29 | 3syl 17 | . . . . . . 7 |
31 | 24, 30 | mpbid 146 | . . . . . 6 |
32 | 31 | a1d 22 | . . . . 5 |
33 | simp3 984 | . . . . . . . . . 10 | |
34 | simp1l 1006 | . . . . . . . . . . 11 | |
35 | simp1r 1007 | . . . . . . . . . . . 12 | |
36 | 35, 25 | syl 14 | . . . . . . . . . . 11 |
37 | ordelsuc 4429 | . . . . . . . . . . 11 | |
38 | 34, 36, 37 | syl2anc 409 | . . . . . . . . . 10 |
39 | 33, 38 | mpbird 166 | . . . . . . . . 9 |
40 | nnsucelsuc 6395 | . . . . . . . . . 10 | |
41 | 35, 40 | syl 14 | . . . . . . . . 9 |
42 | 39, 41 | mpbid 146 | . . . . . . . 8 |
43 | peano2 4517 | . . . . . . . . . 10 | |
44 | 34, 43 | syl 14 | . . . . . . . . 9 |
45 | 36, 26 | syl 14 | . . . . . . . . 9 |
46 | ordelsuc 4429 | . . . . . . . . 9 | |
47 | 44, 45, 46 | syl2anc 409 | . . . . . . . 8 |
48 | 42, 47 | mpbid 146 | . . . . . . 7 |
49 | 48 | 3expia 1184 | . . . . . 6 |
50 | 49 | exp31 362 | . . . . 5 |
51 | 9, 13, 17, 32, 50 | finds2 4523 | . . . 4 |
52 | 5, 51 | vtoclga 2755 | . . 3 |
53 | 52 | imp 123 | . 2 |
54 | nnon 4531 | . . 3 | |
55 | onsucsssucr 4433 | . . 3 | |
56 | 54, 25, 55 | syl2an 287 | . 2 |
57 | 53, 56 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3a 963 wceq 1332 wcel 1481 cvv 2689 wss 3076 c0 3368 word 4292 con0 4293 csuc 4295 com 4512 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-tr 4035 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 |
This theorem is referenced by: nnaword 6415 ennnfonelemk 11949 ennnfonelemkh 11961 |
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