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Mirrors > Home > NFE 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 SF, 13-Oct-1996.) |
Ref | Expression |
---|---|
sspwb | ⊢ (A ⊆ B ↔ ℘A ⊆ ℘B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 3279 | . . . . 5 ⊢ (x ⊆ A → (A ⊆ B → x ⊆ B)) | |
2 | 1 | com12 27 | . . . 4 ⊢ (A ⊆ B → (x ⊆ A → x ⊆ B)) |
3 | vex 2862 | . . . . 5 ⊢ x ∈ V | |
4 | 3 | elpw 3728 | . . . 4 ⊢ (x ∈ ℘A ↔ x ⊆ A) |
5 | 3 | elpw 3728 | . . . 4 ⊢ (x ∈ ℘B ↔ x ⊆ B) |
6 | 2, 4, 5 | 3imtr4g 261 | . . 3 ⊢ (A ⊆ B → (x ∈ ℘A → x ∈ ℘B)) |
7 | 6 | ssrdv 3278 | . 2 ⊢ (A ⊆ B → ℘A ⊆ ℘B) |
8 | ssel 3267 | . . . 4 ⊢ (℘A ⊆ ℘B → ({x} ∈ ℘A → {x} ∈ ℘B)) | |
9 | snex 4111 | . . . . . 6 ⊢ {x} ∈ V | |
10 | 9 | elpw 3728 | . . . . 5 ⊢ ({x} ∈ ℘A ↔ {x} ⊆ A) |
11 | 3 | snss 3838 | . . . . 5 ⊢ (x ∈ A ↔ {x} ⊆ A) |
12 | 10, 11 | bitr4i 243 | . . . 4 ⊢ ({x} ∈ ℘A ↔ x ∈ A) |
13 | 9 | elpw 3728 | . . . . 5 ⊢ ({x} ∈ ℘B ↔ {x} ⊆ B) |
14 | 3 | snss 3838 | . . . . 5 ⊢ (x ∈ B ↔ {x} ⊆ B) |
15 | 13, 14 | bitr4i 243 | . . . 4 ⊢ ({x} ∈ ℘B ↔ x ∈ B) |
16 | 8, 12, 15 | 3imtr3g 260 | . . 3 ⊢ (℘A ⊆ ℘B → (x ∈ A → x ∈ B)) |
17 | 16 | ssrdv 3278 | . 2 ⊢ (℘A ⊆ ℘B → A ⊆ B) |
18 | 7, 17 | impbii 180 | 1 ⊢ (A ⊆ B ↔ ℘A ⊆ ℘B) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∈ wcel 1710 ⊆ wss 3257 ℘cpw 3722 {csn 3737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 |
This theorem is referenced by: pw1ss 4169 sfinltfin 4535 ce2le 6233 |
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