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Theorem sspwb 3943
 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 2946 . . . . 5 (xA → (ABxB))
21com12 27 . . . 4 (AB → (xAxB))
3 vex 2554 . . . . 5 x V
43elpw 3357 . . . 4 (x 𝒫 AxA)
53elpw 3357 . . . 4 (x 𝒫 BxB)
62, 4, 53imtr4g 194 . . 3 (AB → (x 𝒫 Ax 𝒫 B))
76ssrdv 2945 . 2 (AB → 𝒫 A ⊆ 𝒫 B)
8 ssel 2933 . . . 4 (𝒫 A ⊆ 𝒫 B → ({x} 𝒫 A → {x} 𝒫 B))
9 snexgOLD 3926 . . . . . . 7 (x V → {x} V)
103, 9ax-mp 7 . . . . . 6 {x} V
1110elpw 3357 . . . . 5 ({x} 𝒫 A ↔ {x} ⊆ A)
123snss 3485 . . . . 5 (x A ↔ {x} ⊆ A)
1311, 12bitr4i 176 . . . 4 ({x} 𝒫 Ax A)
1410elpw 3357 . . . . 5 ({x} 𝒫 B ↔ {x} ⊆ B)
153snss 3485 . . . . 5 (x B ↔ {x} ⊆ B)
1614, 15bitr4i 176 . . . 4 ({x} 𝒫 Bx B)
178, 13, 163imtr3g 193 . . 3 (𝒫 A ⊆ 𝒫 B → (x Ax B))
1817ssrdv 2945 . 2 (𝒫 A ⊆ 𝒫 BAB)
197, 18impbii 117 1 (AB ↔ 𝒫 A ⊆ 𝒫 B)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911  𝒫 cpw 3351  {csn 3367 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373 This theorem is referenced by:  pwel  3945  ssextss  3947  pweqb  3950
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