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Mirrors > Home > ILE Home > Th. List > sspwb | GIF version |
Description: Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.) |
Ref | Expression |
---|---|
sspwb | ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 3154 | . . . . 5 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵)) | |
2 | 1 | com12 30 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵)) |
3 | vex 2733 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | 3 | elpw 3570 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
5 | 3 | elpw 3570 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) |
6 | 2, 4, 5 | 3imtr4g 204 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)) |
7 | 6 | ssrdv 3153 | . 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
8 | ssel 3141 | . . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵)) | |
9 | 3 | snex 4169 | . . . . . 6 ⊢ {𝑥} ∈ V |
10 | 9 | elpw 3570 | . . . . 5 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴) |
11 | 3 | snss 3707 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴) |
12 | 10, 11 | bitr4i 186 | . . . 4 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ 𝑥 ∈ 𝐴) |
13 | 9 | elpw 3570 | . . . . 5 ⊢ ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵) |
14 | 3 | snss 3707 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ {𝑥} ⊆ 𝐵) |
15 | 13, 14 | bitr4i 186 | . . . 4 ⊢ ({𝑥} ∈ 𝒫 𝐵 ↔ 𝑥 ∈ 𝐵) |
16 | 8, 12, 15 | 3imtr3g 203 | . . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
17 | 16 | ssrdv 3153 | . 2 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → 𝐴 ⊆ 𝐵) |
18 | 7, 17 | impbii 125 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 2141 ⊆ wss 3121 𝒫 cpw 3564 {csn 3581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 |
This theorem is referenced by: pwel 4201 ssextss 4203 pweqb 4206 fiss 6950 pw1on 7190 ntrss 12872 |
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