Theorem ordpwsucss 4477

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 4288 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 4349) and U. { x e. On | x C_ A } = A (onuniss2 4423).

Constructively ( ~P A i^i On ) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4480). (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 4473	. . . . 5 |- ( Ord A <-> Ord suc A )
2		ordelon 4300	. . . . . 6 |- ( ( Ord suc A /\ x e. suc A ) -> x e. On )
3	2	ex 114	. . . . 5 |- ( Ord suc A -> ( x e. suc A -> x e. On ) )
4	1, 3	sylbi 120	. . . 4 |- ( Ord A -> ( x e. suc A -> x e. On ) )
5		ordtr 4295	. . . . 5 |- ( Ord A -> Tr A )
6		trsucss 4340	. . . . 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 305	. . 3 |- ( Ord A -> ( x e. suc A -> ( x e. On /\ x C_ A ) ) )
9		elin 3254	. . . 4 |- ( x e. ( ~P A i^i On ) <-> ( x e. ~P A /\ x e. On ) )
10		velpw 3512	. . . . 5 |- ( x e. ~P A <-> x C_ A )
11	10	anbi2ci 454	. . . 4 |- ( ( x e. ~P A /\ x e. On ) <-> ( x e. On /\ x C_ A ) )
12	9, 11	bitri 183	. . 3 |- ( x e. ( ~P A i^i On ) <-> ( x e. On /\ x C_ A ) )
13	8, 12	syl6ibr 161	. 2 |- ( Ord A -> ( x e. suc A -> x e. ( ~P A i^i On ) ) )
14	13	ssrdv 3098	1 |- ( Ord A -> suc A C_ ( ~P A i^i On ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   i^i cin 3065   C_ wss 3066   ~Pcpw 3505   Tr wtr 4021   Ord word 4279   Oncon0 4280   suc csuc 4282

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 2119  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288

This theorem is referenced by: (None)