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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 4300 | . . 3 | |
2 | suctr 4343 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | df-suc 4293 | . . . . . 6 | |
5 | 4 | eleq2i 2206 | . . . . 5 |
6 | elun 3217 | . . . . 5 | |
7 | velsn 3544 | . . . . . 6 | |
8 | 7 | orbi2i 751 | . . . . 5 |
9 | 5, 6, 8 | 3bitri 205 | . . . 4 |
10 | dford3 4289 | . . . . . . . 8 | |
11 | 10 | simprbi 273 | . . . . . . 7 |
12 | df-ral 2421 | . . . . . . 7 | |
13 | 11, 12 | sylib 121 | . . . . . 6 |
14 | 13 | 19.21bi 1537 | . . . . 5 |
15 | treq 4032 | . . . . . 6 | |
16 | 1, 15 | syl5ibrcom 156 | . . . . 5 |
17 | 14, 16 | jaod 706 | . . . 4 |
18 | 9, 17 | syl5bi 151 | . . 3 |
19 | 18 | ralrimiv 2504 | . 2 |
20 | dford3 4289 | . 2 | |
21 | 3, 19, 20 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 697 wal 1329 wceq 1331 wcel 1480 wral 2416 cun 3069 csn 3527 wtr 4026 word 4284 csuc 4287 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-uni 3737 df-tr 4027 df-iord 4288 df-suc 4293 |
This theorem is referenced by: suceloni 4417 ordsucg 4418 onsucsssucr 4425 ordtriexmidlem 4435 2ordpr 4439 ordsuc 4478 nnsucsssuc 6388 |
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