Theorem ordpwsucss 4291

Description: The collection of ordinals in the power class of an ordinal is a superset of its successor.

We can think of ( ~P A i^i On ) as another possible definition of successor, which would be equivalent to df-suc 4108 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if A e. On then both U. suc A = A (onunisuci 4169) and U. { x e. On | x C_ A } = A (onuniss2 4238).

Constructively ( ~P A i^i On ) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4294). (Contributed by Jim Kingdon, 21-Jul-2019.)

Assertion

Ref	Expression
ordpwsucss	|- ( Ord A -> suc A C_ ( ~P A i^i On ) )

Proof of Theorem ordpwsucss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordsuc 4287	. . . . 5 |- ( Ord A <-> Ord suc A )
2		ordelon 4120	. . . . . 6 |- ( ( Ord suc A /\ x e. suc A ) -> x e. On )
3	2	ex 108	. . . . 5 |- ( Ord suc A -> ( x e. suc A -> x e. On ) )
4	1, 3	sylbi 114	. . . 4 |- ( Ord A -> ( x e. suc A -> x e. On ) )
5		ordtr 4115	. . . . 5 |- ( Ord A -> Tr A )
6		trsucss 4160	. . . . 5 |- ( Tr A -> ( x e. suc A -> x C_ A ) )
7	5, 6	syl 14	. . . 4 |- ( Ord A -> ( x e. suc A -> x C_ A ) )
8	4, 7	jcad 291	. . 3 |- ( Ord A -> ( x e. suc A -> ( x e. On /\ x C_ A ) ) )
9		elin 3126	. . . 4 |- ( x e. ( ~P A i^i On ) <-> ( x e. ~P A /\ x e. On ) )
10		selpw 3366	. . . . 5 |- ( x e. ~P A <-> x C_ A )
11	10	anbi2ci 432	. . . 4 |- ( ( x e. ~P A /\ x e. On ) <-> ( x e. On /\ x C_ A ) )
12	9, 11	bitri 173	. . 3 |- ( x e. ( ~P A i^i On ) <-> ( x e. On /\ x C_ A ) )
13	8, 12	syl6ibr 151	. 2 |- ( Ord A -> ( x e. suc A -> x e. ( ~P A i^i On ) ) )
14	13	ssrdv 2951	1 |- ( Ord A -> suc A C_ ( ~P A i^i On ) )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   i^i cin 2916   C_ wss 2917   ~Pcpw 3359   Tr wtr 3854   Ord word 4099   Oncon0 4100   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108

This theorem is referenced by: (None)