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Theorem sspwb 4006
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 3017 . . . . 5 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
21com12 30 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 vex 2615 . . . . 5 𝑥 ∈ V
43elpw 3412 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53elpw 3412 . . . 4 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
62, 4, 53imtr4g 203 . . 3 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
76ssrdv 3016 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
8 ssel 3004 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵))
93snex 3983 . . . . . 6 {𝑥} ∈ V
109elpw 3412 . . . . 5 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
113snss 3540 . . . . 5 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
1210, 11bitr4i 185 . . . 4 ({𝑥} ∈ 𝒫 𝐴𝑥𝐴)
139elpw 3412 . . . . 5 ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵)
143snss 3540 . . . . 5 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
1513, 14bitr4i 185 . . . 4 ({𝑥} ∈ 𝒫 𝐵𝑥𝐵)
168, 12, 153imtr3g 202 . . 3 (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥𝐴𝑥𝐵))
1716ssrdv 3016 . 2 (𝒫 𝐴 ⊆ 𝒫 𝐵𝐴𝐵)
187, 17impbii 124 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 103  wcel 1434  wss 2984  𝒫 cpw 3406  {csn 3422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428
This theorem is referenced by:  pwel  4008  ssextss  4010  pweqb  4013
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