Theorem pw1on 7338

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 6515	. . . . . 6 |- 1o = { (/) }
2		elsni 3651	. . . . . . . 8 |- ( x e. { (/) } -> x = (/) )
3		0elpw 4208	. . . . . . . 8 |- (/) e. ~P 1o
4	2, 3	eqeltrdi 2296	. . . . . . 7 |- ( x e. { (/) } -> x e. ~P 1o )
5	4	ssriv 3197	. . . . . 6 |- { (/) } C_ ~P 1o
6	1, 5	eqsstri 3225	. . . . 5 |- 1o C_ ~P 1o
7		sspwb 4260	. . . . 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 4147	. . . 4 |- ( Tr ~P 1o <-> ~P 1o C_ ~P ~P 1o )
10	8, 9	mpbir 146	. . 3 |- Tr ~P 1o
11		elpwi 3625	. . . . . . . . 9 |- ( x e. ~P 1o -> x C_ 1o )
12	11	sselda 3193	. . . . . . . 8 |- ( ( x e. ~P 1o /\ y e. x ) -> y e. 1o )
13		el1o 6523	. . . . . . . 8 |- ( y e. 1o <-> y = (/) )
14	12, 13	sylib 122	. . . . . . 7 |- ( ( x e. ~P 1o /\ y e. x ) -> y = (/) )
15		0ss 3499	. . . . . . 7 |- (/) C_ x
16	14, 15	eqsstrdi 3245	. . . . . 6 |- ( ( x e. ~P 1o /\ y e. x ) -> y C_ x )
17	16	ralrimiva 2579	. . . . 5 |- ( x e. ~P 1o -> A. y e. x y C_ x )
18		dftr3 4146	. . . . 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 2559	. . 3 |- A. x e. ~P 1o Tr x
21		dford3 4414	. . 3 |- ( Ord ~P 1o <-> ( Tr ~P 1o /\ A. x e. ~P 1o Tr x ) )
22	10, 20, 21	mpbir2an 945	. 2 |- Ord ~P 1o
23		1oex 6510	. . 3 |- 1o e. _V
24	23	pwex 4227	. 2 |- ~P 1o e. _V
25		elon2 4423	. 2 |- ( ~P 1o e. On <-> ( Ord ~P 1o /\ ~P 1o e. _V ) )
26	22, 24, 25	mpbir2an 945	1 |- ~P 1o e. On

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2176   A.wral 2484   _Vcvv 2772   C_ wss 3166   (/)c0 3460   ~Pcpw 3616   {csn 3633   Tr wtr 4142   Ord word 4409   Oncon0 4410   1oc1o 6495

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415  df-suc 4418  df-1o 6502

This theorem is referenced by:  pw1ne1  7341  sucpw1nss3  7347  onntri35  7349  onntri45  7353