Theorem pw1on 7155

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 6373	. . . . . 6 |- 1o = { (/) }
2		elsni 3578	. . . . . . . 8 |- ( x e. { (/) } -> x = (/) )
3		0elpw 4125	. . . . . . . 8 |- (/) e. ~P 1o
4	2, 3	eqeltrdi 2248	. . . . . . 7 |- ( x e. { (/) } -> x e. ~P 1o )
5	4	ssriv 3132	. . . . . 6 |- { (/) } C_ ~P 1o
6	1, 5	eqsstri 3160	. . . . 5 |- 1o C_ ~P 1o
7		sspwb 4176	. . . . 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 4067	. . . 4 |- ( Tr ~P 1o <-> ~P 1o C_ ~P ~P 1o )
10	8, 9	mpbir 145	. . 3 |- Tr ~P 1o
11		elpwi 3552	. . . . . . . . 9 |- ( x e. ~P 1o -> x C_ 1o )
12	11	sselda 3128	. . . . . . . 8 |- ( ( x e. ~P 1o /\ y e. x ) -> y e. 1o )
13		el1o 6381	. . . . . . . 8 |- ( y e. 1o <-> y = (/) )
14	12, 13	sylib 121	. . . . . . 7 |- ( ( x e. ~P 1o /\ y e. x ) -> y = (/) )
15		0ss 3432	. . . . . . 7 |- (/) C_ x
16	14, 15	eqsstrdi 3180	. . . . . 6 |- ( ( x e. ~P 1o /\ y e. x ) -> y C_ x )
17	16	ralrimiva 2530	. . . . 5 |- ( x e. ~P 1o -> A. y e. x y C_ x )
18		dftr3 4066	. . . . 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 2510	. . 3 |- A. x e. ~P 1o Tr x
21		dford3 4327	. . 3 |- ( Ord ~P 1o <-> ( Tr ~P 1o /\ A. x e. ~P 1o Tr x ) )
22	10, 20, 21	mpbir2an 927	. 2 |- Ord ~P 1o
23		1oex 6368	. . 3 |- 1o e. _V
24	23	pwex 4144	. 2 |- ~P 1o e. _V
25		elon2 4336	. 2 |- ( ~P 1o e. On <-> ( Ord ~P 1o /\ ~P 1o e. _V ) )
26	22, 24, 25	mpbir2an 927	1 |- ~P 1o e. On

Syntax hints:   /\ wa 103   = wceq 1335   e. wcel 2128   A.wral 2435   _Vcvv 2712   C_ wss 3102   (/)c0 3394   ~Pcpw 3543   {csn 3560   Tr wtr 4062   Ord word 4322   Oncon0 4323   1oc1o 6353

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331  df-1o 6360

This theorem is referenced by:  pw1ne1  7158  sucpw1nss3  7164  onntri35  7166  onntri45  7170