Theorem pw1on 7182

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 6397	. . . . . 6 |- 1o = { (/) }
2		elsni 3594	. . . . . . . 8 |- ( x e. { (/) } -> x = (/) )
3		0elpw 4143	. . . . . . . 8 |- (/) e. ~P 1o
4	2, 3	eqeltrdi 2257	. . . . . . 7 |- ( x e. { (/) } -> x e. ~P 1o )
5	4	ssriv 3146	. . . . . 6 |- { (/) } C_ ~P 1o
6	1, 5	eqsstri 3174	. . . . 5 |- 1o C_ ~P 1o
7		sspwb 4194	. . . . 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 4085	. . . 4 |- ( Tr ~P 1o <-> ~P 1o C_ ~P ~P 1o )
10	8, 9	mpbir 145	. . 3 |- Tr ~P 1o
11		elpwi 3568	. . . . . . . . 9 |- ( x e. ~P 1o -> x C_ 1o )
12	11	sselda 3142	. . . . . . . 8 |- ( ( x e. ~P 1o /\ y e. x ) -> y e. 1o )
13		el1o 6405	. . . . . . . 8 |- ( y e. 1o <-> y = (/) )
14	12, 13	sylib 121	. . . . . . 7 |- ( ( x e. ~P 1o /\ y e. x ) -> y = (/) )
15		0ss 3447	. . . . . . 7 |- (/) C_ x
16	14, 15	eqsstrdi 3194	. . . . . 6 |- ( ( x e. ~P 1o /\ y e. x ) -> y C_ x )
17	16	ralrimiva 2539	. . . . 5 |- ( x e. ~P 1o -> A. y e. x y C_ x )
18		dftr3 4084	. . . . 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 2519	. . 3 |- A. x e. ~P 1o Tr x
21		dford3 4345	. . 3 |- ( Ord ~P 1o <-> ( Tr ~P 1o /\ A. x e. ~P 1o Tr x ) )
22	10, 20, 21	mpbir2an 932	. 2 |- Ord ~P 1o
23		1oex 6392	. . 3 |- 1o e. _V
24	23	pwex 4162	. 2 |- ~P 1o e. _V
25		elon2 4354	. 2 |- ( ~P 1o e. On <-> ( Ord ~P 1o /\ ~P 1o e. _V ) )
26	22, 24, 25	mpbir2an 932	1 |- ~P 1o e. On

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   _Vcvv 2726   C_ wss 3116   (/)c0 3409   ~Pcpw 3559   {csn 3576   Tr wtr 4080   Ord word 4340   Oncon0 4341   1oc1o 6377

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349  df-1o 6384

This theorem is referenced by:  pw1ne1  7185  sucpw1nss3  7191  onntri35  7193  onntri45  7197