Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sspwb | Structured version Visualization version 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 3610 | . . . . 5 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵)) | |
2 | 1 | com12 32 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵)) |
3 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | 3 | elpw 4164 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
5 | 3 | elpw 4164 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) |
6 | 2, 4, 5 | 3imtr4g 285 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)) |
7 | 6 | ssrdv 3609 | . 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
8 | ssel 3597 | . . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵)) | |
9 | snex 4908 | . . . . . 6 ⊢ {𝑥} ∈ V | |
10 | 9 | elpw 4164 | . . . . 5 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴) |
11 | 3 | snss 4316 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴) |
12 | 10, 11 | bitr4i 267 | . . . 4 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ 𝑥 ∈ 𝐴) |
13 | 9 | elpw 4164 | . . . . 5 ⊢ ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵) |
14 | 3 | snss 4316 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ {𝑥} ⊆ 𝐵) |
15 | 13, 14 | bitr4i 267 | . . . 4 ⊢ ({𝑥} ∈ 𝒫 𝐵 ↔ 𝑥 ∈ 𝐵) |
16 | 8, 12, 15 | 3imtr3g 284 | . . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
17 | 16 | ssrdv 3609 | . 2 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → 𝐴 ⊆ 𝐵) |
18 | 7, 17 | impbii 199 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∈ wcel 1990 ⊆ wss 3574 𝒫 cpw 4158 {csn 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-sn 4178 df-pr 4180 |
This theorem is referenced by: pwel 4920 ssextss 4922 pweqb 4925 pwdom 8112 marypha1lem 8339 wdompwdom 8483 r1pwss 8647 pwwf 8670 rankpwi 8686 rankxplim 8742 ackbij2lem1 9041 fictb 9067 ssfin2 9142 ssfin3ds 9152 ttukeylem2 9332 hashbclem 13236 wrdexg 13315 incexclem 14568 hashbcss 15708 isacs1i 16318 mreacs 16319 acsfn 16320 sscpwex 16475 wunfunc 16559 isacs3lem 17166 isacs5lem 17169 tgcmp 21204 imastopn 21523 fgabs 21683 fgtr 21694 trfg 21695 ssufl 21722 alexsubb 21850 tsmsres 21947 cfiluweak 22099 cfilresi 23093 cmetss 23113 minveclem4a 23201 minveclem4 23203 vitali 23382 sqff1o 24908 elsigagen2 30211 ldsysgenld 30223 ldgenpisyslem1 30226 measres 30285 imambfm 30324 ballotlem2 30550 neibastop1 32354 neibastop2lem 32355 neibastop2 32356 sstotbnd2 33573 isnacs3 37273 aomclem2 37625 dssmapnvod 38314 gneispace 38432 sge0less 40609 sge0iunmptlemre 40632 |
Copyright terms: Public domain | W3C validator |