Theorem pw1on 7443

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 6595	. . . . . 6 |- 1o = { (/) }
2		elsni 3687	. . . . . . . 8 |- ( x e. { (/) } -> x = (/) )
3		0elpw 4254	. . . . . . . 8 |- (/) e. ~P 1o
4	2, 3	eqeltrdi 2322	. . . . . . 7 |- ( x e. { (/) } -> x e. ~P 1o )
5	4	ssriv 3231	. . . . . 6 |- { (/) } C_ ~P 1o
6	1, 5	eqsstri 3259	. . . . 5 |- 1o C_ ~P 1o
7		sspwb 4308	. . . . 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 4192	. . . 4 |- ( Tr ~P 1o <-> ~P 1o C_ ~P ~P 1o )
10	8, 9	mpbir 146	. . 3 |- Tr ~P 1o
11		elpwi 3661	. . . . . . . . 9 |- ( x e. ~P 1o -> x C_ 1o )
12	11	sselda 3227	. . . . . . . 8 |- ( ( x e. ~P 1o /\ y e. x ) -> y e. 1o )
13		el1o 6604	. . . . . . . 8 |- ( y e. 1o <-> y = (/) )
14	12, 13	sylib 122	. . . . . . 7 |- ( ( x e. ~P 1o /\ y e. x ) -> y = (/) )
15		0ss 3533	. . . . . . 7 |- (/) C_ x
16	14, 15	eqsstrdi 3279	. . . . . 6 |- ( ( x e. ~P 1o /\ y e. x ) -> y C_ x )
17	16	ralrimiva 2605	. . . . 5 |- ( x e. ~P 1o -> A. y e. x y C_ x )
18		dftr3 4191	. . . . 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 2585	. . 3 |- A. x e. ~P 1o Tr x
21		dford3 4464	. . 3 |- ( Ord ~P 1o <-> ( Tr ~P 1o /\ A. x e. ~P 1o Tr x ) )
22	10, 20, 21	mpbir2an 950	. 2 |- Ord ~P 1o
23		1oex 6589	. . 3 |- 1o e. _V
24	23	pwex 4273	. 2 |- ~P 1o e. _V
25		elon2 4473	. 2 |- ( ~P 1o e. On <-> ( Ord ~P 1o /\ ~P 1o e. _V ) )
26	22, 24, 25	mpbir2an 950	1 |- ~P 1o e. On

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   (/)c0 3494   ~Pcpw 3652   {csn 3669   Tr wtr 4187   Ord word 4459   Oncon0 4460   1oc1o 6574

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468  df-1o 6581

This theorem is referenced by:  pw1ne1  7446  sucpw1nss3  7452  onntri35  7454  onntri45  7458