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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 4419, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4440. (Contributed by Jim Kingdon, 25-Aug-2019.) |
Ref | Expression |
---|---|
nnsucelsuc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2201 | . . . 4 | |
2 | suceq 4319 | . . . . 5 | |
3 | 2 | eleq2d 2207 | . . . 4 |
4 | 1, 3 | imbi12d 233 | . . 3 |
5 | eleq2 2201 | . . . 4 | |
6 | suceq 4319 | . . . . 5 | |
7 | 6 | eleq2d 2207 | . . . 4 |
8 | 5, 7 | imbi12d 233 | . . 3 |
9 | eleq2 2201 | . . . 4 | |
10 | suceq 4319 | . . . . 5 | |
11 | 10 | eleq2d 2207 | . . . 4 |
12 | 9, 11 | imbi12d 233 | . . 3 |
13 | eleq2 2201 | . . . 4 | |
14 | suceq 4319 | . . . . 5 | |
15 | 14 | eleq2d 2207 | . . . 4 |
16 | 13, 15 | imbi12d 233 | . . 3 |
17 | noel 3362 | . . . 4 | |
18 | 17 | pm2.21i 635 | . . 3 |
19 | elsuci 4320 | . . . . . . . 8 | |
20 | 19 | adantl 275 | . . . . . . 7 |
21 | simpl 108 | . . . . . . . 8 | |
22 | suceq 4319 | . . . . . . . . 9 | |
23 | 22 | a1i 9 | . . . . . . . 8 |
24 | 21, 23 | orim12d 775 | . . . . . . 7 |
25 | 20, 24 | mpd 13 | . . . . . 6 |
26 | vex 2684 | . . . . . . . 8 | |
27 | 26 | sucex 4410 | . . . . . . 7 |
28 | 27 | elsuc2 4324 | . . . . . 6 |
29 | 25, 28 | sylibr 133 | . . . . 5 |
30 | 29 | ex 114 | . . . 4 |
31 | 30 | a1i 9 | . . 3 |
32 | 4, 8, 12, 16, 18, 31 | finds 4509 | . 2 |
33 | nnon 4518 | . . 3 | |
34 | onsucelsucr 4419 | . . 3 | |
35 | 33, 34 | syl 14 | . 2 |
36 | 32, 35 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 c0 3358 con0 4280 csuc 4282 com 4499 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-tr 4022 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 |
This theorem is referenced by: nnsucsssuc 6381 nntri3or 6382 nnsucuniel 6384 nnaordi 6397 ennnfonelemhom 11917 |
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