Theorem pw1on 7374

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 6540	. . . . . 6 ⊢ 1o = {∅}
2		elsni 3662	. . . . . . . 8 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
3		0elpw 4225	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 1o
4	2, 3	eqeltrdi 2298	. . . . . . 7 ⊢ (𝑥 ∈ {∅} → 𝑥 ∈ 𝒫 1o)
5	4	ssriv 3206	. . . . . 6 ⊢ {∅} ⊆ 𝒫 1o
6	1, 5	eqsstri 3234	. . . . 5 ⊢ 1o ⊆ 𝒫 1o
7		sspwb 4279	. . . . 5 ⊢ (1o ⊆ 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o)
8	6, 7	mpbi 145	. . . 4 ⊢ 𝒫 1o ⊆ 𝒫 𝒫 1o
9		dftr4 4164	. . . 4 ⊢ (Tr 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o)
10	8, 9	mpbir 146	. . 3 ⊢ Tr 𝒫 1o
11		elpwi 3636	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 1o → 𝑥 ⊆ 1o)
12	11	sselda 3202	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 1o)
13		el1o 6548	. . . . . . . 8 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅)
14	12, 13	sylib 122	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 = ∅)
15		0ss 3508	. . . . . . 7 ⊢ ∅ ⊆ 𝑥
16	14, 15	eqsstrdi 3254	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ 𝑥)
17	16	ralrimiva 2581	. . . . 5 ⊢ (𝑥 ∈ 𝒫 1o → ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
18		dftr3 4163	. . . . 5 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
19	17, 18	sylibr 134	. . . 4 ⊢ (𝑥 ∈ 𝒫 1o → Tr 𝑥)
20	19	rgen 2561	. . 3 ⊢ ∀𝑥 ∈ 𝒫 1oTr 𝑥
21		dford3 4433	. . 3 ⊢ (Ord 𝒫 1o ↔ (Tr 𝒫 1o ∧ ∀𝑥 ∈ 𝒫 1oTr 𝑥))
22	10, 20, 21	mpbir2an 945	. 2 ⊢ Ord 𝒫 1o
23		1oex 6535	. . 3 ⊢ 1o ∈ V
24	23	pwex 4244	. 2 ⊢ 𝒫 1o ∈ V
25		elon2 4442	. 2 ⊢ (𝒫 1o ∈ On ↔ (Ord 𝒫 1o ∧ 𝒫 1o ∈ V))
26	22, 24, 25	mpbir2an 945	1 ⊢ 𝒫 1o ∈ On

Syntax hints:   ∧ wa 104   = wceq 1373   ∈ wcel 2178  ∀wral 2486  Vcvv 2777   ⊆ wss 3175  ∅c0 3469  𝒫 cpw 3627  {csn 3644  Tr wtr 4159  Ord word 4428  Oncon0 4429  1_oc1o 6520

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-nul 4187  ax-pow 4235  ax-pr 4270  ax-un 4499

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2779  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-nul 3470  df-pw 3629  df-sn 3650  df-pr 3651  df-uni 3866  df-tr 4160  df-iord 4432  df-on 4434  df-suc 4437  df-1o 6527

This theorem is referenced by:  pw1ne1  7377  sucpw1nss3  7383  onntri35  7385  onntri45  7389