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Mirrors > Home > ILE Home > Th. List > nnsucelsuc | Unicode version |
Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4252, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4273. (Contributed by Jim Kingdon, 25-Aug-2019.) |
Ref | Expression |
---|---|
nnsucelsuc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2142 | . . . 4 | |
2 | suceq 4157 | . . . . 5 | |
3 | 2 | eleq2d 2148 | . . . 4 |
4 | 1, 3 | imbi12d 232 | . . 3 |
5 | eleq2 2142 | . . . 4 | |
6 | suceq 4157 | . . . . 5 | |
7 | 6 | eleq2d 2148 | . . . 4 |
8 | 5, 7 | imbi12d 232 | . . 3 |
9 | eleq2 2142 | . . . 4 | |
10 | suceq 4157 | . . . . 5 | |
11 | 10 | eleq2d 2148 | . . . 4 |
12 | 9, 11 | imbi12d 232 | . . 3 |
13 | eleq2 2142 | . . . 4 | |
14 | suceq 4157 | . . . . 5 | |
15 | 14 | eleq2d 2148 | . . . 4 |
16 | 13, 15 | imbi12d 232 | . . 3 |
17 | noel 3255 | . . . 4 | |
18 | 17 | pm2.21i 607 | . . 3 |
19 | elsuci 4158 | . . . . . . . 8 | |
20 | 19 | adantl 271 | . . . . . . 7 |
21 | simpl 107 | . . . . . . . 8 | |
22 | suceq 4157 | . . . . . . . . 9 | |
23 | 22 | a1i 9 | . . . . . . . 8 |
24 | 21, 23 | orim12d 732 | . . . . . . 7 |
25 | 20, 24 | mpd 13 | . . . . . 6 |
26 | vex 2604 | . . . . . . . 8 | |
27 | 26 | sucex 4243 | . . . . . . 7 |
28 | 27 | elsuc2 4162 | . . . . . 6 |
29 | 25, 28 | sylibr 132 | . . . . 5 |
30 | 29 | ex 113 | . . . 4 |
31 | 30 | a1i 9 | . . 3 |
32 | 4, 8, 12, 16, 18, 31 | finds 4341 | . 2 |
33 | nnon 4350 | . . 3 | |
34 | onsucelsucr 4252 | . . 3 | |
35 | 33, 34 | syl 14 | . 2 |
36 | 32, 35 | impbid 127 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wo 661 wceq 1284 wcel 1433 c0 3251 con0 4118 csuc 4120 com 4331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 |
This theorem is referenced by: nnsucsssuc 6094 nntri3or 6095 nnaordi 6104 |
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