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Theorem ordpwsucss 4319
 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 4136 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if then both (onunisuci 4197) and (onuniss2 4266). Constructively and cannot be shown to be equivalent (as proved at ordpwsucexmid 4322). (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 4315 . . . . 5
2 ordelon 4148 . . . . . 6
32ex 112 . . . . 5
41, 3sylbi 118 . . . 4
5 ordtr 4143 . . . . 5
6 trsucss 4188 . . . . 5
75, 6syl 14 . . . 4
84, 7jcad 295 . . 3
9 elin 3154 . . . 4
10 selpw 3394 . . . . 5
1110anbi2ci 440 . . . 4
129, 11bitri 177 . . 3
138, 12syl6ibr 155 . 2
1413ssrdv 2979 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101   wcel 1409   cin 2944   wss 2945  cpw 3387   wtr 3882   word 4127  con0 4128   csuc 4130 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4290 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136 This theorem is referenced by: (None)
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