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Mirrors > Home > ILE Home > Th. List > onsucelsucr | Unicode version |
Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4440. However, the converse does hold where is a natural number, as seen at nnsucelsuc 6380. (Contributed by Jim Kingdon, 17-Jul-2019.) |
Ref | Expression |
---|---|
onsucelsucr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2692 | . . . 4 | |
2 | sucexb 4408 | . . . 4 | |
3 | 1, 2 | sylibr 133 | . . 3 |
4 | onelss 4304 | . . . . . . 7 | |
5 | eqimss 3146 | . . . . . . . 8 | |
6 | 5 | a1i 9 | . . . . . . 7 |
7 | 4, 6 | jaod 706 | . . . . . 6 |
8 | 7 | adantl 275 | . . . . 5 |
9 | elsucg 4321 | . . . . . . 7 | |
10 | 2, 9 | sylbi 120 | . . . . . 6 |
11 | 10 | adantr 274 | . . . . 5 |
12 | eloni 4292 | . . . . . 6 | |
13 | ordelsuc 4416 | . . . . . 6 | |
14 | 12, 13 | sylan2 284 | . . . . 5 |
15 | 8, 11, 14 | 3imtr4d 202 | . . . 4 |
16 | 15 | impancom 258 | . . 3 |
17 | 3, 16 | mpancom 418 | . 2 |
18 | 17 | com12 30 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 cvv 2681 wss 3066 word 4279 con0 4280 csuc 4282 |
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-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-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 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-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-tr 4022 df-iord 4283 df-on 4285 df-suc 4288 |
This theorem is referenced by: nnsucelsuc 6380 |
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