Theorem pw1on 7286

Description: The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)

Assertion

Ref	Expression
pw1on	|- ~P 1o e. On

Proof of Theorem pw1on

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df1o2 6482	. . . . . 6 |- 1o = { (/) }
2		elsni 3636	. . . . . . . 8 |- ( x e. { (/) } -> x = (/) )
3		0elpw 4193	. . . . . . . 8 |- (/) e. ~P 1o
4	2, 3	eqeltrdi 2284	. . . . . . 7 |- ( x e. { (/) } -> x e. ~P 1o )
5	4	ssriv 3183	. . . . . 6 |- { (/) } C_ ~P 1o
6	1, 5	eqsstri 3211	. . . . 5 |- 1o C_ ~P 1o
7		sspwb 4245	. . . . 5 |- ( 1o C_ ~P 1o <-> ~P 1o C_ ~P ~P 1o )
8	6, 7	mpbi 145	. . . 4 |- ~P 1o C_ ~P ~P 1o
9		dftr4 4132	. . . 4 |- ( Tr ~P 1o <-> ~P 1o C_ ~P ~P 1o )
10	8, 9	mpbir 146	. . 3 |- Tr ~P 1o
11		elpwi 3610	. . . . . . . . 9 |- ( x e. ~P 1o -> x C_ 1o )
12	11	sselda 3179	. . . . . . . 8 |- ( ( x e. ~P 1o /\ y e. x ) -> y e. 1o )
13		el1o 6490	. . . . . . . 8 |- ( y e. 1o <-> y = (/) )
14	12, 13	sylib 122	. . . . . . 7 |- ( ( x e. ~P 1o /\ y e. x ) -> y = (/) )
15		0ss 3485	. . . . . . 7 |- (/) C_ x
16	14, 15	eqsstrdi 3231	. . . . . 6 |- ( ( x e. ~P 1o /\ y e. x ) -> y C_ x )
17	16	ralrimiva 2567	. . . . 5 |- ( x e. ~P 1o -> A. y e. x y C_ x )
18		dftr3 4131	. . . . 5 |- ( Tr x <-> A. y e. x y C_ x )
19	17, 18	sylibr 134	. . . 4 |- ( x e. ~P 1o -> Tr x )
20	19	rgen 2547	. . 3 |- A. x e. ~P 1o Tr x
21		dford3 4398	. . 3 |- ( Ord ~P 1o <-> ( Tr ~P 1o /\ A. x e. ~P 1o Tr x ) )
22	10, 20, 21	mpbir2an 944	. 2 |- Ord ~P 1o
23		1oex 6477	. . 3 |- 1o e. _V
24	23	pwex 4212	. 2 |- ~P 1o e. _V
25		elon2 4407	. 2 |- ( ~P 1o e. On <-> ( Ord ~P 1o /\ ~P 1o e. _V ) )
26	22, 24, 25	mpbir2an 944	1 |- ~P 1o e. On

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   _Vcvv 2760   C_ wss 3153   (/)c0 3446   ~Pcpw 3601   {csn 3618   Tr wtr 4127   Ord word 4393   Oncon0 4394   1oc1o 6462

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402  df-1o 6469

This theorem is referenced by:  pw1ne1  7289  sucpw1nss3  7295  onntri35  7297  onntri45  7301