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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 4273. However, the converse does hold where is a natural number, as seen at nnsucelsuc 6093. (Contributed by Jim Kingdon, 17-Jul-2019.) |
Ref | Expression |
---|---|
onsucelsucr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2610 | . . . 4 | |
2 | sucexb 4241 | . . . 4 | |
3 | 1, 2 | sylibr 132 | . . 3 |
4 | onelss 4142 | . . . . . . 7 | |
5 | eqimss 3051 | . . . . . . . 8 | |
6 | 5 | a1i 9 | . . . . . . 7 |
7 | 4, 6 | jaod 669 | . . . . . 6 |
8 | 7 | adantl 271 | . . . . 5 |
9 | elsucg 4159 | . . . . . . 7 | |
10 | 2, 9 | sylbi 119 | . . . . . 6 |
11 | 10 | adantr 270 | . . . . 5 |
12 | eloni 4130 | . . . . . 6 | |
13 | ordelsuc 4249 | . . . . . 6 | |
14 | 12, 13 | sylan2 280 | . . . . 5 |
15 | 8, 11, 14 | 3imtr4d 201 | . . . 4 |
16 | 15 | impancom 256 | . . 3 |
17 | 3, 16 | mpancom 413 | . 2 |
18 | 17 | com12 30 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wo 661 wceq 1284 wcel 1433 cvv 2601 wss 2973 word 4117 con0 4118 csuc 4120 |
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-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-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 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-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 |
This theorem is referenced by: nnsucelsuc 6093 |
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