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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 4354 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if then both (onunisuci 4415) and (onuniss2 4494). Constructively and cannot be shown to be equivalent (as proved at ordpwsucexmid 4552). (Contributed by Jim Kingdon, 21-Jul-2019.) |
Ref | Expression |
---|---|
ordpwsucss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsuc 4545 | . . . . 5 | |
2 | ordelon 4366 | . . . . . 6 | |
3 | 2 | ex 114 | . . . . 5 |
4 | 1, 3 | sylbi 120 | . . . 4 |
5 | ordtr 4361 | . . . . 5 | |
6 | trsucss 4406 | . . . . 5 | |
7 | 5, 6 | syl 14 | . . . 4 |
8 | 4, 7 | jcad 305 | . . 3 |
9 | elin 3310 | . . . 4 | |
10 | velpw 3571 | . . . . 5 | |
11 | 10 | anbi2ci 456 | . . . 4 |
12 | 9, 11 | bitri 183 | . . 3 |
13 | 8, 12 | syl6ibr 161 | . 2 |
14 | 13 | ssrdv 3153 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2141 cin 3120 wss 3121 cpw 3564 wtr 4085 word 4345 con0 4346 csuc 4348 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-setind 4519 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-uni 3795 df-tr 4086 df-iord 4349 df-on 4351 df-suc 4354 |
This theorem is referenced by: (None) |
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