Theorem ordpwsuc 3896

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

Assertion

Ref	Expression
ordpwsuc	|- (Ord A -> (~PA i^i On) = suc A)

Proof of Theorem ordpwsuc

Step	Hyp	Ref	Expression
1		ordsssuc 3756	. . . . . 6 |- ((x e. On /\ Ord A) -> (x C_ A <-> x e. suc A))
2	1	expcom 403	. . . . 5 |- (Ord A -> (x e. On -> (x C_ A <-> x e. suc A)))
3	2	pm5.32d 709	. . . 4 |- (Ord A -> ((x e. On /\ x C_ A) <-> (x e. On /\ x e. suc A)))
4		simpr 350	. . . . 5 |- ((x e. On /\ x e. suc A) -> x e. suc A)
5		ordsuc 3895	. . . . . . 7 |- (Ord A <-> Ord suc A)
6		ordelon 3682	. . . . . . . 8 |- ((Ord suc A /\ x e. suc A) -> x e. On)
7	6	ex 402	. . . . . . 7 |- (Ord suc A -> (x e. suc A -> x e. On))
8	5, 7	sylbi 216	. . . . . 6 |- (Ord A -> (x e. suc A -> x e. On))
9	8	ancrd 323	. . . . 5 |- (Ord A -> (x e. suc A -> (x e. On /\ x e. suc A)))
10	4, 9	impbid2 576	. . . 4 |- (Ord A -> ((x e. On /\ x e. suc A) <-> x e. suc A))
11	3, 10	bitrd 587	. . 3 |- (Ord A -> ((x e. On /\ x C_ A) <-> x e. suc A))
12		elin 2786	. . . 4 |- (x e. (~PA i^i On) <-> (x e. ~PA /\ x e. On))
13		visset 2295	. . . . . 6 |- x e. _V
14	13	elpw 3037	. . . . 5 |- (x e. ~PA <-> x C_ A)
15	14	anbi1i 539	. . . 4 |- ((x e. ~PA /\ x e. On) <-> (x C_ A /\ x e. On))
16		ancom 482	. . . 4 |- ((x C_ A /\ x e. On) <-> (x e. On /\ x C_ A))
17	12, 15, 16	3bitri 194	. . 3 |- (x e. (~PA i^i On) <-> (x e. On /\ x C_ A))
18	11, 17	syl5bb 591	. 2 |- (Ord A -> (x e. (~PA i^i On) <-> x e. suc A))
19	18	eqrdv 1882	1 |- (Ord A -> (~PA i^i On) = suc A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   i^i cin 2592   C_ wss 2593  ~Pcpw 3032  Ord word 3656  Oncon0 3657  suc csuc 3659

This theorem is referenced by:  onpwsuc 3897  orduniss2 3913

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663

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