Theorem ordpwsuc 3066

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 3057	. . . . . 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 647	. . . 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 3065	. . . . . . 7 |- (Ord A <-> Ord suc A)
6		ordelon 2971	. . . . . . . 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 518	. . . 4 |- (Ord A -> ((x e. On /\ x e. suc A) <-> x e. suc A))
11	3, 10	bitrd 528	. . 3 |- (Ord A -> ((x e. On /\ x (_ A) <-> x e. suc A))
12		elin 2207	. . . 4 |- (x e. (P~A i^i On) <-> (x e. P~A /\ x e. On))
13		visset 1813	. . . . . 6 |- x e. V
14	13	elpw 2404	. . . . 5 |- (x e. P~A <-> x (_ A)
15	14	anbi1i 481	. . . 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 532	. 2 |- (Ord A -> (x e. (P~A i^i On) <-> x e. suc A))
19	18	eqrdv 1473	1 |- (Ord A -> (P~A i^i On) = suc A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958   i^i cin 2046   (_ wss 2047  P~cpw 2401  Ord word 2947  Oncon0 2948  suc csuc 2950

This theorem is referenced by:  onpwsuc 3067  orduniss2 3090

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954

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