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Mirrors > Home > ILE Home > Th. List > suctr | Unicode version |
Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) |
Ref | Expression |
---|---|
suctr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . 5 | |
2 | vex 2715 | . . . . . 6 | |
3 | 2 | elsuc 4366 | . . . . 5 |
4 | 1, 3 | sylib 121 | . . . 4 |
5 | simpl 108 | . . . . . . 7 | |
6 | eleq2 2221 | . . . . . . 7 | |
7 | 5, 6 | syl5ibcom 154 | . . . . . 6 |
8 | elelsuc 4369 | . . . . . 6 | |
9 | 7, 8 | syl6 33 | . . . . 5 |
10 | trel 4069 | . . . . . . . . 9 | |
11 | 10 | expd 256 | . . . . . . . 8 |
12 | 11 | adantrd 277 | . . . . . . 7 |
13 | 12, 8 | syl8 71 | . . . . . 6 |
14 | jao 745 | . . . . . 6 | |
15 | 13, 14 | syl6 33 | . . . . 5 |
16 | 9, 15 | mpdi 43 | . . . 4 |
17 | 4, 16 | mpdi 43 | . . 3 |
18 | 17 | alrimivv 1855 | . 2 |
19 | dftr2 4064 | . 2 | |
20 | 18, 19 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wal 1333 wceq 1335 wcel 2128 wtr 4062 csuc 4325 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-uni 3773 df-tr 4063 df-suc 4331 |
This theorem is referenced by: ordsucim 4458 ordom 4565 |
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