Theorem ordpwsucss 4604

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 4407 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 4468) and U. { x e. On | x C_ A } = A (onuniss2 4549).

Constructively ( ~P A i^i On ) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4607). (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 4600	. . . . 5 |- ( Ord A <-> Ord suc A )
2		ordelon 4419	. . . . . 6 |- ( ( Ord suc A /\ x e. suc A ) -> x e. On )
3	2	ex 115	. . . . 5 |- ( Ord suc A -> ( x e. suc A -> x e. On ) )
4	1, 3	sylbi 121	. . . 4 |- ( Ord A -> ( x e. suc A -> x e. On ) )
5		ordtr 4414	. . . . 5 |- ( Ord A -> Tr A )
6		trsucss 4459	. . . . 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 307	. . 3 |- ( Ord A -> ( x e. suc A -> ( x e. On /\ x C_ A ) ) )
9		elin 3347	. . . 4 |- ( x e. ( ~P A i^i On ) <-> ( x e. ~P A /\ x e. On ) )
10		velpw 3613	. . . . 5 |- ( x e. ~P A <-> x C_ A )
11	10	anbi2ci 459	. . . 4 |- ( ( x e. ~P A /\ x e. On ) <-> ( x e. On /\ x C_ A ) )
12	9, 11	bitri 184	. . 3 |- ( x e. ( ~P A i^i On ) <-> ( x e. On /\ x C_ A ) )
13	8, 12	imbitrrdi 162	. 2 |- ( Ord A -> ( x e. suc A -> x e. ( ~P A i^i On ) ) )
14	13	ssrdv 3190	1 |- ( Ord A -> suc A C_ ( ~P A i^i On ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   i^i cin 3156   C_ wss 3157   ~Pcpw 3606   Tr wtr 4132   Ord word 4398   Oncon0 4399   suc csuc 4401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-uni 3841  df-tr 4133  df-iord 4402  df-on 4404  df-suc 4407

This theorem is referenced by: (None)