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Mirrors > Home > ILE Home > Th. List > ordsucim | Unicode version |
Description: The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.) |
Ref | Expression |
---|---|
ordsucim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 4270 | . . 3 | |
2 | suctr 4313 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | df-suc 4263 | . . . . . 6 | |
5 | 4 | eleq2i 2184 | . . . . 5 |
6 | elun 3187 | . . . . 5 | |
7 | velsn 3514 | . . . . . 6 | |
8 | 7 | orbi2i 736 | . . . . 5 |
9 | 5, 6, 8 | 3bitri 205 | . . . 4 |
10 | dford3 4259 | . . . . . . . 8 | |
11 | 10 | simprbi 273 | . . . . . . 7 |
12 | df-ral 2398 | . . . . . . 7 | |
13 | 11, 12 | sylib 121 | . . . . . 6 |
14 | 13 | 19.21bi 1522 | . . . . 5 |
15 | treq 4002 | . . . . . 6 | |
16 | 1, 15 | syl5ibrcom 156 | . . . . 5 |
17 | 14, 16 | jaod 691 | . . . 4 |
18 | 9, 17 | syl5bi 151 | . . 3 |
19 | 18 | ralrimiv 2481 | . 2 |
20 | dford3 4259 | . 2 | |
21 | 3, 19, 20 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 682 wal 1314 wceq 1316 wcel 1465 wral 2393 cun 3039 csn 3497 wtr 3996 word 4254 csuc 4257 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-uni 3707 df-tr 3997 df-iord 4258 df-suc 4263 |
This theorem is referenced by: suceloni 4387 ordsucg 4388 onsucsssucr 4395 ordtriexmidlem 4405 2ordpr 4409 ordsuc 4448 nnsucsssuc 6356 |
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