Theorem pw1on 7293

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 6487	. . . . . 6 |- 1o = { (/) }
2		elsni 3640	. . . . . . . 8 |- ( x e. { (/) } -> x = (/) )
3		0elpw 4197	. . . . . . . 8 |- (/) e. ~P 1o
4	2, 3	eqeltrdi 2287	. . . . . . 7 |- ( x e. { (/) } -> x e. ~P 1o )
5	4	ssriv 3187	. . . . . 6 |- { (/) } C_ ~P 1o
6	1, 5	eqsstri 3215	. . . . 5 |- 1o C_ ~P 1o
7		sspwb 4249	. . . . 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 4136	. . . 4 |- ( Tr ~P 1o <-> ~P 1o C_ ~P ~P 1o )
10	8, 9	mpbir 146	. . 3 |- Tr ~P 1o
11		elpwi 3614	. . . . . . . . 9 |- ( x e. ~P 1o -> x C_ 1o )
12	11	sselda 3183	. . . . . . . 8 |- ( ( x e. ~P 1o /\ y e. x ) -> y e. 1o )
13		el1o 6495	. . . . . . . 8 |- ( y e. 1o <-> y = (/) )
14	12, 13	sylib 122	. . . . . . 7 |- ( ( x e. ~P 1o /\ y e. x ) -> y = (/) )
15		0ss 3489	. . . . . . 7 |- (/) C_ x
16	14, 15	eqsstrdi 3235	. . . . . 6 |- ( ( x e. ~P 1o /\ y e. x ) -> y C_ x )
17	16	ralrimiva 2570	. . . . 5 |- ( x e. ~P 1o -> A. y e. x y C_ x )
18		dftr3 4135	. . . . 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 2550	. . 3 |- A. x e. ~P 1o Tr x
21		dford3 4402	. . 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 6482	. . 3 |- 1o e. _V
24	23	pwex 4216	. 2 |- ~P 1o e. _V
25		elon2 4411	. 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 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   C_ wss 3157   (/)c0 3450   ~Pcpw 3605   {csn 3622   Tr wtr 4131   Ord word 4397   Oncon0 4398   1oc1o 6467

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403  df-suc 4406  df-1o 6474

This theorem is referenced by:  pw1ne1  7296  sucpw1nss3  7302  onntri35  7304  onntri45  7308