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Theorem sspwb 4947
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.)
Assertion
Ref Expression
sspwb (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sstr2 3643 . . . . 5 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
21com12 32 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 vex 3234 . . . . 5 𝑥 ∈ V
43elpw 4197 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53elpw 4197 . . . 4 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
62, 4, 53imtr4g 285 . . 3 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
76ssrdv 3642 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
8 ssel 3630 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵))
9 snex 4938 . . . . . 6 {𝑥} ∈ V
109elpw 4197 . . . . 5 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
113snss 4348 . . . . 5 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
1210, 11bitr4i 267 . . . 4 ({𝑥} ∈ 𝒫 𝐴𝑥𝐴)
139elpw 4197 . . . . 5 ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵)
143snss 4348 . . . . 5 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
1513, 14bitr4i 267 . . . 4 ({𝑥} ∈ 𝒫 𝐵𝑥𝐵)
168, 12, 153imtr3g 284 . . 3 (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥𝐴𝑥𝐵))
1716ssrdv 3642 . 2 (𝒫 𝐴 ⊆ 𝒫 𝐵𝐴𝐵)
187, 17impbii 199 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2030  wss 3607  𝒫 cpw 4191  {csn 4210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213
This theorem is referenced by:  pwel  4950  ssextss  4952  pweqb  4955  pwdom  8153  marypha1lem  8380  wdompwdom  8524  r1pwss  8685  pwwf  8708  rankpwi  8724  rankxplim  8780  ackbij2lem1  9079  fictb  9105  ssfin2  9180  ssfin3ds  9190  ttukeylem2  9370  hashbclem  13274  wrdexg  13347  incexclem  14612  hashbcss  15755  isacs1i  16365  mreacs  16366  acsfn  16367  sscpwex  16522  wunfunc  16606  isacs3lem  17213  isacs5lem  17216  tgcmp  21252  imastopn  21571  fgabs  21730  fgtr  21741  trfg  21742  ssufl  21769  alexsubb  21897  tsmsres  21994  cfiluweak  22146  cfilresi  23139  cmetss  23159  minveclem4a  23247  minveclem4  23249  vitali  23427  sqff1o  24953  elsigagen2  30339  ldsysgenld  30351  ldgenpisyslem1  30354  measres  30413  imambfm  30452  ballotlem2  30678  neibastop1  32479  neibastop2lem  32480  neibastop2  32481  sstotbnd2  33703  isnacs3  37590  aomclem2  37942  dssmapnvod  38631  gneispace  38749  sge0less  40927  sge0iunmptlemre  40950
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