Theorem pw1on 7221

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 6426	. . . . . 6 |- 1o = { (/) }
2		elsni 3610	. . . . . . . 8 |- ( x e. { (/) } -> x = (/) )
3		0elpw 4163	. . . . . . . 8 |- (/) e. ~P 1o
4	2, 3	eqeltrdi 2268	. . . . . . 7 |- ( x e. { (/) } -> x e. ~P 1o )
5	4	ssriv 3159	. . . . . 6 |- { (/) } C_ ~P 1o
6	1, 5	eqsstri 3187	. . . . 5 |- 1o C_ ~P 1o
7		sspwb 4215	. . . . 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 4105	. . . 4 |- ( Tr ~P 1o <-> ~P 1o C_ ~P ~P 1o )
10	8, 9	mpbir 146	. . 3 |- Tr ~P 1o
11		elpwi 3584	. . . . . . . . 9 |- ( x e. ~P 1o -> x C_ 1o )
12	11	sselda 3155	. . . . . . . 8 |- ( ( x e. ~P 1o /\ y e. x ) -> y e. 1o )
13		el1o 6434	. . . . . . . 8 |- ( y e. 1o <-> y = (/) )
14	12, 13	sylib 122	. . . . . . 7 |- ( ( x e. ~P 1o /\ y e. x ) -> y = (/) )
15		0ss 3461	. . . . . . 7 |- (/) C_ x
16	14, 15	eqsstrdi 3207	. . . . . 6 |- ( ( x e. ~P 1o /\ y e. x ) -> y C_ x )
17	16	ralrimiva 2550	. . . . 5 |- ( x e. ~P 1o -> A. y e. x y C_ x )
18		dftr3 4104	. . . . 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 2530	. . 3 |- A. x e. ~P 1o Tr x
21		dford3 4366	. . 3 |- ( Ord ~P 1o <-> ( Tr ~P 1o /\ A. x e. ~P 1o Tr x ) )
22	10, 20, 21	mpbir2an 942	. 2 |- Ord ~P 1o
23		1oex 6421	. . 3 |- 1o e. _V
24	23	pwex 4182	. 2 |- ~P 1o e. _V
25		elon2 4375	. 2 |- ( ~P 1o e. On <-> ( Ord ~P 1o /\ ~P 1o e. _V ) )
26	22, 24, 25	mpbir2an 942	1 |- ~P 1o e. On

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2737   C_ wss 3129   (/)c0 3422   ~Pcpw 3575   {csn 3592   Tr wtr 4100   Ord word 4361   Oncon0 4362   1oc1o 6406

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4101  df-iord 4365  df-on 4367  df-suc 4370  df-1o 6413

This theorem is referenced by:  pw1ne1  7224  sucpw1nss3  7230  onntri35  7232  onntri45  7236