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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 4501. However, the converse does hold where is a natural number, as seen at nnsucelsuc 6450. (Contributed by Jim Kingdon, 17-Jul-2019.) |
Ref | Expression |
---|---|
onsucelsucr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2732 | . . . 4 | |
2 | sucexb 4468 | . . . 4 | |
3 | 1, 2 | sylibr 133 | . . 3 |
4 | onelss 4359 | . . . . . . 7 | |
5 | eqimss 3191 | . . . . . . . 8 | |
6 | 5 | a1i 9 | . . . . . . 7 |
7 | 4, 6 | jaod 707 | . . . . . 6 |
8 | 7 | adantl 275 | . . . . 5 |
9 | elsucg 4376 | . . . . . . 7 | |
10 | 2, 9 | sylbi 120 | . . . . . 6 |
11 | 10 | adantr 274 | . . . . 5 |
12 | eloni 4347 | . . . . . 6 | |
13 | ordelsuc 4476 | . . . . . 6 | |
14 | 12, 13 | sylan2 284 | . . . . 5 |
15 | 8, 11, 14 | 3imtr4d 202 | . . . 4 |
16 | 15 | impancom 258 | . . 3 |
17 | 3, 16 | mpancom 419 | . 2 |
18 | 17 | com12 30 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wceq 1342 wcel 2135 cvv 2721 wss 3111 word 4334 con0 4335 csuc 4337 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-tr 4075 df-iord 4338 df-on 4340 df-suc 4343 |
This theorem is referenced by: nnsucelsuc 6450 |
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