Theorem ordpwsuc 3056

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

Assertion

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

Proof of Theorem ordpwsuc

Step	Hyp	Ref	Expression
1		ordsssuc 3047	. . . . . 6 |- ((x e. On /\ Ord A) -> (x (_ A <-> x e. suc A))
2	1	expcom 374	. . . . 5 |- (Ord A -> (x e. On -> (x (_ A <-> x e. suc A)))
3	2	pm5.32d 645	. . . 4 |- (Ord A -> ((x e. On /\ x (_ A) <-> (x e. On /\ x e. suc A)))
4		pm3.27 323	. . . . 5 |- ((x e. On /\ x e. suc A) -> x e. suc A)
5		ordsuc 3055	. . . . . . 7 |- (Ord A <-> Ord suc A)
6		ordelon 2961	. . . . . . . 8 |- ((Ord suc A /\ x e. suc A) -> x e. On)
7	6	ex 373	. . . . . . 7 |- (Ord suc A -> (x e. suc A -> x e. On))
8	5, 7	sylbi 199	. . . . . 6 |- (Ord A -> (x e. suc A -> x e. On))
9	8	ancrd 299	. . . . 5 |- (Ord A -> (x e. suc A -> (x e. On /\ x e. suc A)))
10	4, 9	impbid2 516	. . . 4 |- (Ord A -> ((x e. On /\ x e. suc A) <-> x e. suc A))
11	3, 10	bitrd 526	. . 3 |- (Ord A -> ((x e. On /\ x (_ A) <-> x e. suc A))
12		elin 2197	. . . 4 |- (x e. (P~A i^i On) <-> (x e. P~A /\ x e. On))
13		visset 1804	. . . . . 6 |- x e. V
14	13	elpw 2394	. . . . 5 |- (x e. P~A <-> x (_ A)
15	14	anbi1i 480	. . . 4 |- ((x e. P~A /\ x e. On) <-> (x (_ A /\ x e. On))
16		ancom 435	. . . 4 |- ((x (_ A /\ x e. On) <-> (x e. On /\ x (_ A))
17	12, 15, 16	3bitr 177	. . 3 |- (x e. (P~A i^i On) <-> (x e. On /\ x (_ A))
18	11, 17	syl5bb 530	. 2 |- (Ord A -> (x e. (P~A i^i On) <-> x e. suc A))
19	18	eqrdv 1466	1 |- (Ord A -> (P~A i^i On) = suc A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955   i^i cin 2036   (_ wss 2037  P~cpw 2391  Ord word 2937  Oncon0 2938  suc csuc 2940

This theorem is referenced by:  onpwsuc 3057  orduniss2 3080

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944

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