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Theorem sspwb 4138
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 3104 . . . . 5 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
21com12 30 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 vex 2689 . . . . 5 𝑥 ∈ V
43elpw 3516 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53elpw 3516 . . . 4 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
62, 4, 53imtr4g 204 . . 3 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
76ssrdv 3103 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
8 ssel 3091 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵))
93snex 4109 . . . . . 6 {𝑥} ∈ V
109elpw 3516 . . . . 5 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
113snss 3649 . . . . 5 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
1210, 11bitr4i 186 . . . 4 ({𝑥} ∈ 𝒫 𝐴𝑥𝐴)
139elpw 3516 . . . . 5 ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵)
143snss 3649 . . . . 5 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
1513, 14bitr4i 186 . . . 4 ({𝑥} ∈ 𝒫 𝐵𝑥𝐵)
168, 12, 153imtr3g 203 . . 3 (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥𝐴𝑥𝐵))
1716ssrdv 3103 . 2 (𝒫 𝐴 ⊆ 𝒫 𝐵𝐴𝐵)
187, 17impbii 125 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 1480  wss 3071  𝒫 cpw 3510  {csn 3527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533
This theorem is referenced by:  pwel  4140  ssextss  4142  pweqb  4145  fiss  6865  ntrss  12302
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