Theorem pw1on 7549

Description: The power set of 1_o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)

Assertion

Ref	Expression
pw1on	⊢ 𝒫 1o ∈ 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 ∈ {∅} → 𝑥 = ∅)
3		0elpw 4282	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 1o
4	2, 3	eqeltrdi 2325	. . . . . . 7 ⊢ (𝑥 ∈ {∅} → 𝑥 ∈ 𝒫 1o)
5	4	ssriv 3246	. . . . . 6 ⊢ {∅} ⊆ 𝒫 1o
6	1, 5	eqsstri 3274	. . . . 5 ⊢ 1o ⊆ 𝒫 1o
7		sspwb 4337	. . . . 5 ⊢ (1o ⊆ 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o)
8	6, 7	mpbi 145	. . . 4 ⊢ 𝒫 1o ⊆ 𝒫 𝒫 1o
9		dftr4 4218	. . . 4 ⊢ (Tr 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o)
10	8, 9	mpbir 146	. . 3 ⊢ Tr 𝒫 1o
11		elpwi 3683	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 1o → 𝑥 ⊆ 1o)
12	11	sselda 3242	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 1o)
13		el1o 6683	. . . . . . . 8 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅)
14	12, 13	sylib 122	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 = ∅)
15		0ss 3551	. . . . . . 7 ⊢ ∅ ⊆ 𝑥
16	14, 15	eqsstrdi 3294	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ 𝑥)
17	16	ralrimiva 2617	. . . . 5 ⊢ (𝑥 ∈ 𝒫 1o → ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
18		dftr3 4217	. . . . 5 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
19	17, 18	sylibr 134	. . . 4 ⊢ (𝑥 ∈ 𝒫 1o → Tr 𝑥)
20	19	rgen 2597	. . 3 ⊢ ∀𝑥 ∈ 𝒫 1oTr 𝑥
21		dford3 4493	. . 3 ⊢ (Ord 𝒫 1o ↔ (Tr 𝒫 1o ∧ ∀𝑥 ∈ 𝒫 1oTr 𝑥))
22	10, 20, 21	mpbir2an 951	. 2 ⊢ Ord 𝒫 1o
23		1oex 6668	. . 3 ⊢ 1o ∈ V
24	23	pwex 4301	. 2 ⊢ 𝒫 1o ∈ V
25		elon2 4502	. 2 ⊢ (𝒫 1o ∈ On ↔ (Ord 𝒫 1o ∧ 𝒫 1o ∈ V))
26	22, 24, 25	mpbir2an 951	1 ⊢ 𝒫 1o ∈ On

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815   ⊆ wss 3214  ∅c0 3512  𝒫 cpw 3674  {csn 3694  Tr wtr 4213  Ord word 4488  Oncon0 4489  1_oc1o 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