Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4395, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4412. (Contributed by Jim Kingdon, 25-Aug-2019.) |
Ref | Expression |
---|---|
nnsucsssuc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3090 | . . . . . 6 | |
2 | suceq 4294 | . . . . . . 7 | |
3 | 2 | sseq1d 3096 | . . . . . 6 |
4 | 1, 3 | imbi12d 233 | . . . . 5 |
5 | 4 | imbi2d 229 | . . . 4 |
6 | sseq1 3090 | . . . . . 6 | |
7 | suceq 4294 | . . . . . . 7 | |
8 | 7 | sseq1d 3096 | . . . . . 6 |
9 | 6, 8 | imbi12d 233 | . . . . 5 |
10 | sseq1 3090 | . . . . . 6 | |
11 | suceq 4294 | . . . . . . 7 | |
12 | 11 | sseq1d 3096 | . . . . . 6 |
13 | 10, 12 | imbi12d 233 | . . . . 5 |
14 | sseq1 3090 | . . . . . 6 | |
15 | suceq 4294 | . . . . . . 7 | |
16 | 15 | sseq1d 3096 | . . . . . 6 |
17 | 14, 16 | imbi12d 233 | . . . . 5 |
18 | peano3 4480 | . . . . . . . . 9 | |
19 | 18 | neneqd 2306 | . . . . . . . 8 |
20 | peano2 4479 | . . . . . . . . . 10 | |
21 | 0elnn 4502 | . . . . . . . . . 10 | |
22 | 20, 21 | syl 14 | . . . . . . . . 9 |
23 | 22 | ord 698 | . . . . . . . 8 |
24 | 19, 23 | mpd 13 | . . . . . . 7 |
25 | nnord 4495 | . . . . . . . 8 | |
26 | ordsucim 4386 | . . . . . . . 8 | |
27 | 0ex 4025 | . . . . . . . . 9 | |
28 | ordelsuc 4391 | . . . . . . . . 9 | |
29 | 27, 28 | mpan 420 | . . . . . . . 8 |
30 | 25, 26, 29 | 3syl 17 | . . . . . . 7 |
31 | 24, 30 | mpbid 146 | . . . . . 6 |
32 | 31 | a1d 22 | . . . . 5 |
33 | simp3 968 | . . . . . . . . . 10 | |
34 | simp1l 990 | . . . . . . . . . . 11 | |
35 | simp1r 991 | . . . . . . . . . . . 12 | |
36 | 35, 25 | syl 14 | . . . . . . . . . . 11 |
37 | ordelsuc 4391 | . . . . . . . . . . 11 | |
38 | 34, 36, 37 | syl2anc 408 | . . . . . . . . . 10 |
39 | 33, 38 | mpbird 166 | . . . . . . . . 9 |
40 | nnsucelsuc 6355 | . . . . . . . . . 10 | |
41 | 35, 40 | syl 14 | . . . . . . . . 9 |
42 | 39, 41 | mpbid 146 | . . . . . . . 8 |
43 | peano2 4479 | . . . . . . . . . 10 | |
44 | 34, 43 | syl 14 | . . . . . . . . 9 |
45 | 36, 26 | syl 14 | . . . . . . . . 9 |
46 | ordelsuc 4391 | . . . . . . . . 9 | |
47 | 44, 45, 46 | syl2anc 408 | . . . . . . . 8 |
48 | 42, 47 | mpbid 146 | . . . . . . 7 |
49 | 48 | 3expia 1168 | . . . . . 6 |
50 | 49 | exp31 361 | . . . . 5 |
51 | 9, 13, 17, 32, 50 | finds2 4485 | . . . 4 |
52 | 5, 51 | vtoclga 2726 | . . 3 |
53 | 52 | imp 123 | . 2 |
54 | nnon 4493 | . . 3 | |
55 | onsucsssucr 4395 | . . 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 682 w3a 947 wceq 1316 wcel 1465 cvv 2660 wss 3041 c0 3333 word 4254 con0 4255 csuc 4257 com 4474 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 |
This theorem is referenced by: nnaword 6375 ennnfonelemk 11840 ennnfonelemkh 11852 |
Copyright terms: Public domain | W3C validator |