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Mirrors > Home > MPE Home > Th. List > sspwb | Structured version Visualization version GIF version |
Description: The powerclass construction preserves and reflects inclusion. 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 | sspw 4616 | . 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) | |
2 | ssel 3989 | . . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵)) | |
3 | vsnex 5440 | . . . . . 6 ⊢ {𝑥} ∈ V | |
4 | 3 | elpw 4609 | . . . . 5 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴) |
5 | vex 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
6 | 5 | snss 4790 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴) |
7 | 4, 6 | bitr4i 278 | . . . 4 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ 𝑥 ∈ 𝐴) |
8 | 3 | elpw 4609 | . . . . 5 ⊢ ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵) |
9 | 5 | snss 4790 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ {𝑥} ⊆ 𝐵) |
10 | 8, 9 | bitr4i 278 | . . . 4 ⊢ ({𝑥} ∈ 𝒫 𝐵 ↔ 𝑥 ∈ 𝐵) |
11 | 2, 7, 10 | 3imtr3g 295 | . . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
12 | 11 | ssrdv 4001 | . 2 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → 𝐴 ⊆ 𝐵) |
13 | 1, 12 | impbii 209 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ wcel 2106 ⊆ wss 3963 𝒫 cpw 4605 {csn 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 df-pw 4607 df-sn 4632 df-pr 4634 |
This theorem is referenced by: ssextss 5464 pweqb 5467 psspwb 42246 |
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