Theorem pw1on 7293

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 6487	. . . . . 6 ⊢ 1o = {∅}
2		elsni 3640	. . . . . . . 8 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
3		0elpw 4197	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 1o
4	2, 3	eqeltrdi 2287	. . . . . . 7 ⊢ (𝑥 ∈ {∅} → 𝑥 ∈ 𝒫 1o)
5	4	ssriv 3187	. . . . . 6 ⊢ {∅} ⊆ 𝒫 1o
6	1, 5	eqsstri 3215	. . . . 5 ⊢ 1o ⊆ 𝒫 1o
7		sspwb 4249	. . . . 5 ⊢ (1o ⊆ 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o)
8	6, 7	mpbi 145	. . . 4 ⊢ 𝒫 1o ⊆ 𝒫 𝒫 1o
9		dftr4 4136	. . . 4 ⊢ (Tr 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o)
10	8, 9	mpbir 146	. . 3 ⊢ Tr 𝒫 1o
11		elpwi 3614	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 1o → 𝑥 ⊆ 1o)
12	11	sselda 3183	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 1o)
13		el1o 6495	. . . . . . . 8 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅)
14	12, 13	sylib 122	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 = ∅)
15		0ss 3489	. . . . . . 7 ⊢ ∅ ⊆ 𝑥
16	14, 15	eqsstrdi 3235	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ 𝑥)
17	16	ralrimiva 2570	. . . . 5 ⊢ (𝑥 ∈ 𝒫 1o → ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
18		dftr3 4135	. . . . 5 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
19	17, 18	sylibr 134	. . . 4 ⊢ (𝑥 ∈ 𝒫 1o → Tr 𝑥)
20	19	rgen 2550	. . 3 ⊢ ∀𝑥 ∈ 𝒫 1oTr 𝑥
21		dford3 4402	. . 3 ⊢ (Ord 𝒫 1o ↔ (Tr 𝒫 1o ∧ ∀𝑥 ∈ 𝒫 1oTr 𝑥))
22	10, 20, 21	mpbir2an 944	. 2 ⊢ Ord 𝒫 1o
23		1oex 6482	. . 3 ⊢ 1o ∈ V
24	23	pwex 4216	. 2 ⊢ 𝒫 1o ∈ V
25		elon2 4411	. 2 ⊢ (𝒫 1o ∈ On ↔ (Ord 𝒫 1o ∧ 𝒫 1o ∈ V))
26	22, 24, 25	mpbir2an 944	1 ⊢ 𝒫 1o ∈ On

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ⊆ wss 3157  ∅c0 3450  𝒫 cpw 3605  {csn 3622  Tr wtr 4131  Ord word 4397  Oncon0 4398  1_oc1o 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