![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2946 | . . . . 5 ⊢ (x ⊆ A → (A ⊆ B → x ⊆ B)) | |
2 | 1 | com12 27 | . . . 4 ⊢ (A ⊆ B → (x ⊆ A → x ⊆ B)) |
3 | vex 2554 | . . . . 5 ⊢ x ∈ V | |
4 | 3 | elpw 3357 | . . . 4 ⊢ (x ∈ 𝒫 A ↔ x ⊆ A) |
5 | 3 | elpw 3357 | . . . 4 ⊢ (x ∈ 𝒫 B ↔ x ⊆ B) |
6 | 2, 4, 5 | 3imtr4g 194 | . . 3 ⊢ (A ⊆ B → (x ∈ 𝒫 A → x ∈ 𝒫 B)) |
7 | 6 | ssrdv 2945 | . 2 ⊢ (A ⊆ B → 𝒫 A ⊆ 𝒫 B) |
8 | ssel 2933 | . . . 4 ⊢ (𝒫 A ⊆ 𝒫 B → ({x} ∈ 𝒫 A → {x} ∈ 𝒫 B)) | |
9 | snexgOLD 3926 | . . . . . . 7 ⊢ (x ∈ V → {x} ∈ V) | |
10 | 3, 9 | ax-mp 7 | . . . . . 6 ⊢ {x} ∈ V |
11 | 10 | elpw 3357 | . . . . 5 ⊢ ({x} ∈ 𝒫 A ↔ {x} ⊆ A) |
12 | 3 | snss 3485 | . . . . 5 ⊢ (x ∈ A ↔ {x} ⊆ A) |
13 | 11, 12 | bitr4i 176 | . . . 4 ⊢ ({x} ∈ 𝒫 A ↔ x ∈ A) |
14 | 10 | elpw 3357 | . . . . 5 ⊢ ({x} ∈ 𝒫 B ↔ {x} ⊆ B) |
15 | 3 | snss 3485 | . . . . 5 ⊢ (x ∈ B ↔ {x} ⊆ B) |
16 | 14, 15 | bitr4i 176 | . . . 4 ⊢ ({x} ∈ 𝒫 B ↔ x ∈ B) |
17 | 8, 13, 16 | 3imtr3g 193 | . . 3 ⊢ (𝒫 A ⊆ 𝒫 B → (x ∈ A → x ∈ B)) |
18 | 17 | ssrdv 2945 | . 2 ⊢ (𝒫 A ⊆ 𝒫 B → A ⊆ B) |
19 | 7, 18 | impbii 117 | 1 ⊢ (A ⊆ B ↔ 𝒫 A ⊆ 𝒫 B) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∈ wcel 1390 Vcvv 2551 ⊆ wss 2911 𝒫 cpw 3351 {csn 3367 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 |
This theorem is referenced by: pwel 3945 ssextss 3947 pweqb 3950 |
Copyright terms: Public domain | W3C validator |