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Theorem sspwb 4234
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 3177 . . . . 5 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
21com12 30 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 vex 2755 . . . . 5 𝑥 ∈ V
43elpw 3596 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53elpw 3596 . . . 4 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
62, 4, 53imtr4g 205 . . 3 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
76ssrdv 3176 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
8 ssel 3164 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵))
93snex 4203 . . . . . 6 {𝑥} ∈ V
109elpw 3596 . . . . 5 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
113snss 3742 . . . . 5 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
1210, 11bitr4i 187 . . . 4 ({𝑥} ∈ 𝒫 𝐴𝑥𝐴)
139elpw 3596 . . . . 5 ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵)
143snss 3742 . . . . 5 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
1513, 14bitr4i 187 . . . 4 ({𝑥} ∈ 𝒫 𝐵𝑥𝐵)
168, 12, 153imtr3g 204 . . 3 (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥𝐴𝑥𝐵))
1716ssrdv 3176 . 2 (𝒫 𝐴 ⊆ 𝒫 𝐵𝐴𝐵)
187, 17impbii 126 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 105  wcel 2160  wss 3144  𝒫 cpw 3590  {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613
This theorem is referenced by:  pwel  4236  ssextss  4238  pweqb  4241  fiss  7006  pw1on  7255  ntrss  14076
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