Theorem ordpwsucss 4396

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 4207 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 4268) and U. { x e. On | x C_ A } = A (onuniss2 4342).

Constructively ( ~P A i^i On ) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4399). (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 4392	. . . . 5 |- ( Ord A <-> Ord suc A )
2		ordelon 4219	. . . . . 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 4214	. . . . 5 |- ( Ord A -> Tr A )
6		trsucss 4259	. . . . 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 302	. . 3 |- ( Ord A -> ( x e. suc A -> ( x e. On /\ x C_ A ) ) )
9		elin 3184	. . . 4 |- ( x e. ( ~P A i^i On ) <-> ( x e. ~P A /\ x e. On ) )
10		selpw 3440	. . . . 5 |- ( x e. ~P A <-> x C_ A )
11	10	anbi2ci 448	. . . 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 3032	1 |- ( Ord A -> suc A C_ ( ~P A i^i On ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1439   i^i cin 2999   C_ wss 3000   ~Pcpw 3433   Tr wtr 3942   Ord word 4198   Oncon0 4199   suc csuc 4201

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-setind 4366

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-uni 3660  df-tr 3943  df-iord 4202  df-on 4204  df-suc 4207

This theorem is referenced by: (None)