Theorem pw1on 7203

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 6408	. . . . . 6 |- 1o = { (/) }
2		elsni 3601	. . . . . . . 8 |- ( x e. { (/) } -> x = (/) )
3		0elpw 4150	. . . . . . . 8 |- (/) e. ~P 1o
4	2, 3	eqeltrdi 2261	. . . . . . 7 |- ( x e. { (/) } -> x e. ~P 1o )
5	4	ssriv 3151	. . . . . 6 |- { (/) } C_ ~P 1o
6	1, 5	eqsstri 3179	. . . . 5 |- 1o C_ ~P 1o
7		sspwb 4201	. . . . 5 |- ( 1o C_ ~P 1o <-> ~P 1o C_ ~P ~P 1o )
8	6, 7	mpbi 144	. . . 4 |- ~P 1o C_ ~P ~P 1o
9		dftr4 4092	. . . 4 |- ( Tr ~P 1o <-> ~P 1o C_ ~P ~P 1o )
10	8, 9	mpbir 145	. . 3 |- Tr ~P 1o
11		elpwi 3575	. . . . . . . . 9 |- ( x e. ~P 1o -> x C_ 1o )
12	11	sselda 3147	. . . . . . . 8 |- ( ( x e. ~P 1o /\ y e. x ) -> y e. 1o )
13		el1o 6416	. . . . . . . 8 |- ( y e. 1o <-> y = (/) )
14	12, 13	sylib 121	. . . . . . 7 |- ( ( x e. ~P 1o /\ y e. x ) -> y = (/) )
15		0ss 3453	. . . . . . 7 |- (/) C_ x
16	14, 15	eqsstrdi 3199	. . . . . 6 |- ( ( x e. ~P 1o /\ y e. x ) -> y C_ x )
17	16	ralrimiva 2543	. . . . 5 |- ( x e. ~P 1o -> A. y e. x y C_ x )
18		dftr3 4091	. . . . 5 |- ( Tr x <-> A. y e. x y C_ x )
19	17, 18	sylibr 133	. . . 4 |- ( x e. ~P 1o -> Tr x )
20	19	rgen 2523	. . 3 |- A. x e. ~P 1o Tr x
21		dford3 4352	. . 3 |- ( Ord ~P 1o <-> ( Tr ~P 1o /\ A. x e. ~P 1o Tr x ) )
22	10, 20, 21	mpbir2an 937	. 2 |- Ord ~P 1o
23		1oex 6403	. . 3 |- 1o e. _V
24	23	pwex 4169	. 2 |- ~P 1o e. _V
25		elon2 4361	. 2 |- ( ~P 1o e. On <-> ( Ord ~P 1o /\ ~P 1o e. _V ) )
26	22, 24, 25	mpbir2an 937	1 |- ~P 1o e. On

Syntax hints:   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   C_ wss 3121   (/)c0 3414   ~Pcpw 3566   {csn 3583   Tr wtr 4087   Ord word 4347   Oncon0 4348   1oc1o 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356  df-1o 6395

This theorem is referenced by:  pw1ne1  7206  sucpw1nss3  7212  onntri35  7214  onntri45  7218