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Mirrors > Home > ILE Home > Th. List > ordpwsucss | Unicode version |
Description: The collection of
ordinals in the power class of an ordinal is a
superset of its successor.
We can think of as another possible definition of successor, which would be equivalent to df-suc 4263 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if then both (onunisuci 4324) and (onuniss2 4398). Constructively and cannot be shown to be equivalent (as proved at ordpwsucexmid 4455). (Contributed by Jim Kingdon, 21-Jul-2019.) |
Ref | Expression |
---|---|
ordpwsucss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsuc 4448 | . . . . 5 | |
2 | ordelon 4275 | . . . . . 6 | |
3 | 2 | ex 114 | . . . . 5 |
4 | 1, 3 | sylbi 120 | . . . 4 |
5 | ordtr 4270 | . . . . 5 | |
6 | trsucss 4315 | . . . . 5 | |
7 | 5, 6 | syl 14 | . . . 4 |
8 | 4, 7 | jcad 305 | . . 3 |
9 | elin 3229 | . . . 4 | |
10 | velpw 3487 | . . . . 5 | |
11 | 10 | anbi2ci 454 | . . . 4 |
12 | 9, 11 | bitri 183 | . . 3 |
13 | 8, 12 | syl6ibr 161 | . 2 |
14 | 13 | ssrdv 3073 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 cin 3040 wss 3041 cpw 3480 wtr 3996 word 4254 con0 4255 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-in1 588 ax-in2 589 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 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 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-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 |
This theorem is referenced by: (None) |
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