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Theorem sspwb 3979
 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 2979 . . . . 5 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
21com12 30 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 vex 2577 . . . . 5 𝑥 ∈ V
43elpw 3392 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53elpw 3392 . . . 4 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
62, 4, 53imtr4g 198 . . 3 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
76ssrdv 2978 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
8 ssel 2966 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵))
9 snexgOLD 3962 . . . . . . 7 (𝑥 ∈ V → {𝑥} ∈ V)
103, 9ax-mp 7 . . . . . 6 {𝑥} ∈ V
1110elpw 3392 . . . . 5 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
123snss 3521 . . . . 5 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
1311, 12bitr4i 180 . . . 4 ({𝑥} ∈ 𝒫 𝐴𝑥𝐴)
1410elpw 3392 . . . . 5 ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵)
153snss 3521 . . . . 5 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
1614, 15bitr4i 180 . . . 4 ({𝑥} ∈ 𝒫 𝐵𝑥𝐵)
178, 13, 163imtr3g 197 . . 3 (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥𝐴𝑥𝐵))
1817ssrdv 2978 . 2 (𝒫 𝐴 ⊆ 𝒫 𝐵𝐴𝐵)
197, 18impbii 121 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 102   ∈ wcel 1409  Vcvv 2574   ⊆ wss 2944  𝒫 cpw 3386  {csn 3402 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408 This theorem is referenced by:  pwel  3981  ssextss  3983  pweqb  3986
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