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Mirrors > Home > ILE Home > Th. List > pw1on | GIF version |
Description: The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.) |
Ref | Expression |
---|---|
pw1on | ⊢ 𝒫 1o ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 6424 | . . . . . 6 ⊢ 1o = {∅} | |
2 | elsni 3609 | . . . . . . . 8 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅) | |
3 | 0elpw 4161 | . . . . . . . 8 ⊢ ∅ ∈ 𝒫 1o | |
4 | 2, 3 | eqeltrdi 2268 | . . . . . . 7 ⊢ (𝑥 ∈ {∅} → 𝑥 ∈ 𝒫 1o) |
5 | 4 | ssriv 3159 | . . . . . 6 ⊢ {∅} ⊆ 𝒫 1o |
6 | 1, 5 | eqsstri 3187 | . . . . 5 ⊢ 1o ⊆ 𝒫 1o |
7 | sspwb 4213 | . . . . 5 ⊢ (1o ⊆ 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o) | |
8 | 6, 7 | mpbi 145 | . . . 4 ⊢ 𝒫 1o ⊆ 𝒫 𝒫 1o |
9 | dftr4 4103 | . . . 4 ⊢ (Tr 𝒫 1o ↔ 𝒫 1o ⊆ 𝒫 𝒫 1o) | |
10 | 8, 9 | mpbir 146 | . . 3 ⊢ Tr 𝒫 1o |
11 | elpwi 3583 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 1o → 𝑥 ⊆ 1o) | |
12 | 11 | sselda 3155 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 1o) |
13 | el1o 6432 | . . . . . . . 8 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅) | |
14 | 12, 13 | sylib 122 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 = ∅) |
15 | 0ss 3461 | . . . . . . 7 ⊢ ∅ ⊆ 𝑥 | |
16 | 14, 15 | eqsstrdi 3207 | . . . . . 6 ⊢ ((𝑥 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ 𝑥) |
17 | 16 | ralrimiva 2550 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 1o → ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥) |
18 | dftr3 4102 | . . . . 5 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥) | |
19 | 17, 18 | sylibr 134 | . . . 4 ⊢ (𝑥 ∈ 𝒫 1o → Tr 𝑥) |
20 | 19 | rgen 2530 | . . 3 ⊢ ∀𝑥 ∈ 𝒫 1oTr 𝑥 |
21 | dford3 4364 | . . 3 ⊢ (Ord 𝒫 1o ↔ (Tr 𝒫 1o ∧ ∀𝑥 ∈ 𝒫 1oTr 𝑥)) | |
22 | 10, 20, 21 | mpbir2an 942 | . 2 ⊢ Ord 𝒫 1o |
23 | 1oex 6419 | . . 3 ⊢ 1o ∈ V | |
24 | 23 | pwex 4180 | . 2 ⊢ 𝒫 1o ∈ V |
25 | elon2 4373 | . 2 ⊢ (𝒫 1o ∈ On ↔ (Ord 𝒫 1o ∧ 𝒫 1o ∈ V)) | |
26 | 22, 24, 25 | mpbir2an 942 | 1 ⊢ 𝒫 1o ∈ On |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 Vcvv 2737 ⊆ wss 3129 ∅c0 3422 𝒫 cpw 3574 {csn 3591 Tr wtr 4098 Ord word 4359 Oncon0 4360 1oc1o 6404 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-uni 3808 df-tr 4099 df-iord 4363 df-on 4365 df-suc 4368 df-1o 6411 |
This theorem is referenced by: pw1ne1 7222 sucpw1nss3 7228 onntri35 7230 onntri45 7234 |
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