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Mirrors > Home > ILE Home > Th. List > trsuc | Unicode version |
Description: A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
trsuc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sssucid 4152 | . . . . . 6 | |
2 | ssexg 3896 | . . . . . 6 | |
3 | 1, 2 | mpan 400 | . . . . 5 |
4 | sucidg 4153 | . . . . 5 | |
5 | 3, 4 | syl 14 | . . . 4 |
6 | 5 | ancri 307 | . . 3 |
7 | trel 3861 | . . 3 | |
8 | 6, 7 | syl5 28 | . 2 |
9 | 8 | imp 115 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wcel 1393 cvv 2557 wss 2917 wtr 3854 csuc 4102 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-uni 3581 df-tr 3855 df-suc 4108 |
This theorem is referenced by: (None) |
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