Theorem ordpwsucss 4310

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 4126 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 4187) and U. { x e. On | x C_ A } = A (onuniss2 4256).

Constructively ( ~P A i^i On ) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4313). (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 4306	. . . . 5 |- ( Ord A <-> Ord suc A )
2		ordelon 4138	. . . . . 6 |- ( ( Ord suc A /\ x e. suc A ) -> x e. On )
3	2	ex 113	. . . . 5 |- ( Ord suc A -> ( x e. suc A -> x e. On ) )
4	1, 3	sylbi 119	. . . 4 |- ( Ord A -> ( x e. suc A -> x e. On ) )
5		ordtr 4133	. . . . 5 |- ( Ord A -> Tr A )
6		trsucss 4178	. . . . 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 301	. . 3 |- ( Ord A -> ( x e. suc A -> ( x e. On /\ x C_ A ) ) )
9		elin 3155	. . . 4 |- ( x e. ( ~P A i^i On ) <-> ( x e. ~P A /\ x e. On ) )
10		selpw 3389	. . . . 5 |- ( x e. ~P A <-> x C_ A )
11	10	anbi2ci 446	. . . 4 |- ( ( x e. ~P A /\ x e. On ) <-> ( x e. On /\ x C_ A ) )
12	9, 11	bitri 182	. . 3 |- ( x e. ( ~P A i^i On ) <-> ( x e. On /\ x C_ A ) )
13	8, 12	syl6ibr 160	. 2 |- ( Ord A -> ( x e. suc A -> x e. ( ~P A i^i On ) ) )
14	13	ssrdv 3005	1 |- ( Ord A -> suc A C_ ( ~P A i^i On ) )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   i^i cin 2972   C_ wss 2973   ~Pcpw 3382   Tr wtr 3875   Ord word 4117   Oncon0 4118   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126

This theorem is referenced by: (None)