Theorem pw1on 7337

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 6514	. . . . . 6 |- 1o = { (/) }
2		elsni 3650	. . . . . . . 8 |- ( x e. { (/) } -> x = (/) )
3		0elpw 4207	. . . . . . . 8 |- (/) e. ~P 1o
4	2, 3	eqeltrdi 2295	. . . . . . 7 |- ( x e. { (/) } -> x e. ~P 1o )
5	4	ssriv 3196	. . . . . 6 |- { (/) } C_ ~P 1o
6	1, 5	eqsstri 3224	. . . . 5 |- 1o C_ ~P 1o
7		sspwb 4259	. . . . 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 4146	. . . 4 |- ( Tr ~P 1o <-> ~P 1o C_ ~P ~P 1o )
10	8, 9	mpbir 146	. . 3 |- Tr ~P 1o
11		elpwi 3624	. . . . . . . . 9 |- ( x e. ~P 1o -> x C_ 1o )
12	11	sselda 3192	. . . . . . . 8 |- ( ( x e. ~P 1o /\ y e. x ) -> y e. 1o )
13		el1o 6522	. . . . . . . 8 |- ( y e. 1o <-> y = (/) )
14	12, 13	sylib 122	. . . . . . 7 |- ( ( x e. ~P 1o /\ y e. x ) -> y = (/) )
15		0ss 3498	. . . . . . 7 |- (/) C_ x
16	14, 15	eqsstrdi 3244	. . . . . 6 |- ( ( x e. ~P 1o /\ y e. x ) -> y C_ x )
17	16	ralrimiva 2578	. . . . 5 |- ( x e. ~P 1o -> A. y e. x y C_ x )
18		dftr3 4145	. . . . 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 2558	. . 3 |- A. x e. ~P 1o Tr x
21		dford3 4413	. . 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 6509	. . 3 |- 1o e. _V
24	23	pwex 4226	. 2 |- ~P 1o e. _V
25		elon2 4422	. 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 1372   e. wcel 2175   A.wral 2483   _Vcvv 2771   C_ wss 3165   (/)c0 3459   ~Pcpw 3615   {csn 3632   Tr wtr 4141   Ord word 4408   Oncon0 4409   1oc1o 6494

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-uni 3850  df-tr 4142  df-iord 4412  df-on 4414  df-suc 4417  df-1o 6501

This theorem is referenced by:  pw1ne1  7340  sucpw1nss3  7346  onntri35  7348  onntri45  7352