Theorem pw1on 7434

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 6591	. . . . . 6 |- 1o = { (/) }
2		elsni 3685	. . . . . . . 8 |- ( x e. { (/) } -> x = (/) )
3		0elpw 4252	. . . . . . . 8 |- (/) e. ~P 1o
4	2, 3	eqeltrdi 2320	. . . . . . 7 |- ( x e. { (/) } -> x e. ~P 1o )
5	4	ssriv 3229	. . . . . 6 |- { (/) } C_ ~P 1o
6	1, 5	eqsstri 3257	. . . . 5 |- 1o C_ ~P 1o
7		sspwb 4306	. . . . 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 4190	. . . 4 |- ( Tr ~P 1o <-> ~P 1o C_ ~P ~P 1o )
10	8, 9	mpbir 146	. . 3 |- Tr ~P 1o
11		elpwi 3659	. . . . . . . . 9 |- ( x e. ~P 1o -> x C_ 1o )
12	11	sselda 3225	. . . . . . . 8 |- ( ( x e. ~P 1o /\ y e. x ) -> y e. 1o )
13		el1o 6600	. . . . . . . 8 |- ( y e. 1o <-> y = (/) )
14	12, 13	sylib 122	. . . . . . 7 |- ( ( x e. ~P 1o /\ y e. x ) -> y = (/) )
15		0ss 3531	. . . . . . 7 |- (/) C_ x
16	14, 15	eqsstrdi 3277	. . . . . 6 |- ( ( x e. ~P 1o /\ y e. x ) -> y C_ x )
17	16	ralrimiva 2603	. . . . 5 |- ( x e. ~P 1o -> A. y e. x y C_ x )
18		dftr3 4189	. . . . 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 2583	. . 3 |- A. x e. ~P 1o Tr x
21		dford3 4462	. . 3 |- ( Ord ~P 1o <-> ( Tr ~P 1o /\ A. x e. ~P 1o Tr x ) )
22	10, 20, 21	mpbir2an 948	. 2 |- Ord ~P 1o
23		1oex 6585	. . 3 |- 1o e. _V
24	23	pwex 4271	. 2 |- ~P 1o e. _V
25		elon2 4471	. 2 |- ( ~P 1o e. On <-> ( Ord ~P 1o /\ ~P 1o e. _V ) )
26	22, 24, 25	mpbir2an 948	1 |- ~P 1o e. On

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2800   C_ wss 3198   (/)c0 3492   ~Pcpw 3650   {csn 3667   Tr wtr 4185   Ord word 4457   Oncon0 4458   1oc1o 6570

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-uni 3892  df-tr 4186  df-iord 4461  df-on 4463  df-suc 4466  df-1o 6577

This theorem is referenced by:  pw1ne1  7437  sucpw1nss3  7443  onntri35  7445  onntri45  7449