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Theorem ordpwsucss 4482
 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 4293 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if then both (onunisuci 4354) and (onuniss2 4428). Constructively and cannot be shown to be equivalent (as proved at ordpwsucexmid 4485). (Contributed by Jim Kingdon, 21-Jul-2019.)
Assertion
Ref Expression
ordpwsucss

Proof of Theorem ordpwsucss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ordsuc 4478 . . . . 5
2 ordelon 4305 . . . . . 6
32ex 114 . . . . 5
41, 3sylbi 120 . . . 4
5 ordtr 4300 . . . . 5
6 trsucss 4345 . . . . 5
75, 6syl 14 . . . 4
84, 7jcad 305 . . 3
9 elin 3259 . . . 4
10 velpw 3517 . . . . 5
1110anbi2ci 454 . . . 4
129, 11bitri 183 . . 3
138, 12syl6ibr 161 . 2
1413ssrdv 3103 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wcel 1480   cin 3070   wss 3071  cpw 3510   wtr 4026   word 4284  con0 4285   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-in1 603  ax-in2 604  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  ax-setind 4452 This theorem depends on definitions:  df-bi 116  df-3an 964  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-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293 This theorem is referenced by: (None)
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