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Theorem sspwb 4118
 Description: Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by SF, 13-Oct-1996.)
Assertion
Ref Expression
sspwb (A BA B)

Proof of Theorem sspwb
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sstr2 3279 . . . . 5 (x A → (A Bx B))
21com12 27 . . . 4 (A B → (x Ax B))
3 vex 2862 . . . . 5 x V
43elpw 3728 . . . 4 (x Ax A)
53elpw 3728 . . . 4 (x Bx B)
62, 4, 53imtr4g 261 . . 3 (A B → (x Ax B))
76ssrdv 3278 . 2 (A BA B)
8 ssel 3267 . . . 4 (A B → ({x} A → {x} B))
9 snex 4111 . . . . . 6 {x} V
109elpw 3728 . . . . 5 ({x} A ↔ {x} A)
113snss 3838 . . . . 5 (x A ↔ {x} A)
1210, 11bitr4i 243 . . . 4 ({x} Ax A)
139elpw 3728 . . . . 5 ({x} B ↔ {x} B)
143snss 3838 . . . . 5 (x B ↔ {x} B)
1513, 14bitr4i 243 . . . 4 ({x} Bx B)
168, 12, 153imtr3g 260 . . 3 (A B → (x Ax B))
1716ssrdv 3278 . 2 (A BA B)
187, 17impbii 180 1 (A BA B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∈ wcel 1710   ⊆ wss 3257  ℘cpw 3722  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741 This theorem is referenced by:  pw1ss  4169  sfinltfin  4535  ce2le  6233
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