Theorem pw1on 7399

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 6565	. . . . . 6 ⊢ 1o = {∅}
2		elsni 3684	. . . . . . . 8 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
3		0elpw 4247	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 1o
4	2, 3	eqeltrdi 2320	. . . . . . 7 ⊢ (𝑥 ∈ {∅} → 𝑥 ∈ 𝒫 1o)
5	4	ssriv 3228	. . . . . 6 ⊢ {∅} ⊆ 𝒫 1o
6	1, 5	eqsstri 3256	. . . . 5 ⊢ 1o ⊆ 𝒫 1o
7		sspwb 4301	. . . . 5 ⊢ (1o ⊆ 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o)
8	6, 7	mpbi 145	. . . 4 ⊢ 𝒫 1o ⊆ 𝒫 𝒫 1o
9		dftr4 4186	. . . 4 ⊢ (Tr 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o)
10	8, 9	mpbir 146	. . 3 ⊢ Tr 𝒫 1o
11		elpwi 3658	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 1o → 𝑥 ⊆ 1o)
12	11	sselda 3224	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 1o)
13		el1o 6573	. . . . . . . 8 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅)
14	12, 13	sylib 122	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 = ∅)
15		0ss 3530	. . . . . . 7 ⊢ ∅ ⊆ 𝑥
16	14, 15	eqsstrdi 3276	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ 𝑥)
17	16	ralrimiva 2603	. . . . 5 ⊢ (𝑥 ∈ 𝒫 1o → ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
18		dftr3 4185	. . . . 5 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
19	17, 18	sylibr 134	. . . 4 ⊢ (𝑥 ∈ 𝒫 1o → Tr 𝑥)
20	19	rgen 2583	. . 3 ⊢ ∀𝑥 ∈ 𝒫 1oTr 𝑥
21		dford3 4455	. . 3 ⊢ (Ord 𝒫 1o ↔ (Tr 𝒫 1o ∧ ∀𝑥 ∈ 𝒫 1oTr 𝑥))
22	10, 20, 21	mpbir2an 948	. 2 ⊢ Ord 𝒫 1o
23		1oex 6560	. . 3 ⊢ 1o ∈ V
24	23	pwex 4266	. 2 ⊢ 𝒫 1o ∈ V
25		elon2 4464	. 2 ⊢ (𝒫 1o ∈ On ↔ (Ord 𝒫 1o ∧ 𝒫 1o ∈ V))
26	22, 24, 25	mpbir2an 948	1 ⊢ 𝒫 1o ∈ On

Syntax hints:   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∀wral 2508  Vcvv 2799   ⊆ wss 3197  ∅c0 3491  𝒫 cpw 3649  {csn 3666  Tr wtr 4181  Ord word 4450  Oncon0 4451  1_oc1o 6545

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 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521

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 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-tr 4182  df-iord 4454  df-on 4456  df-suc 4459  df-1o 6552

This theorem is referenced by:  pw1ne1  7402  sucpw1nss3  7408  onntri35  7410  onntri45  7414