Theorem pw1on 7549

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 6674	. . . . . 6 |- 1o = { (/) }
2		elsni 3712	. . . . . . . 8 |- ( x e. { (/) } -> x = (/) )
3		0elpw 4282	. . . . . . . 8 |- (/) e. ~P 1o
4	2, 3	eqeltrdi 2325	. . . . . . 7 |- ( x e. { (/) } -> x e. ~P 1o )
5	4	ssriv 3246	. . . . . 6 |- { (/) } C_ ~P 1o
6	1, 5	eqsstri 3274	. . . . 5 |- 1o C_ ~P 1o
7		sspwb 4337	. . . . 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 4218	. . . 4 |- ( Tr ~P 1o <-> ~P 1o C_ ~P ~P 1o )
10	8, 9	mpbir 146	. . 3 |- Tr ~P 1o
11		elpwi 3683	. . . . . . . . 9 |- ( x e. ~P 1o -> x C_ 1o )
12	11	sselda 3242	. . . . . . . 8 |- ( ( x e. ~P 1o /\ y e. x ) -> y e. 1o )
13		el1o 6683	. . . . . . . 8 |- ( y e. 1o <-> y = (/) )
14	12, 13	sylib 122	. . . . . . 7 |- ( ( x e. ~P 1o /\ y e. x ) -> y = (/) )
15		0ss 3551	. . . . . . 7 |- (/) C_ x
16	14, 15	eqsstrdi 3294	. . . . . 6 |- ( ( x e. ~P 1o /\ y e. x ) -> y C_ x )
17	16	ralrimiva 2617	. . . . 5 |- ( x e. ~P 1o -> A. y e. x y C_ x )
18		dftr3 4217	. . . . 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 2597	. . 3 |- A. x e. ~P 1o Tr x
21		dford3 4493	. . 3 |- ( Ord ~P 1o <-> ( Tr ~P 1o /\ A. x e. ~P 1o Tr x ) )
22	10, 20, 21	mpbir2an 951	. 2 |- Ord ~P 1o
23		1oex 6668	. . 3 |- 1o e. _V
24	23	pwex 4301	. 2 |- ~P 1o e. _V
25		elon2 4502	. 2 |- ( ~P 1o e. On <-> ( Ord ~P 1o /\ ~P 1o e. _V ) )
26	22, 24, 25	mpbir2an 951	1 |- ~P 1o e. On

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   C_ wss 3214   (/)c0 3512   ~Pcpw 3674   {csn 3694   Tr wtr 4213   Ord word 4488   Oncon0 4489   1oc1o 6653

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-tr 4214  df-iord 4492  df-on 4494  df-suc 4497  df-1o 6660

This theorem is referenced by:  pw1ne1  7552  sucpw1nss3  7558  onntri35  7560  onntri45  7564