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Theorem ssextss 4205
Description: An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
Assertion
Ref Expression
ssextss (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssextss
StepHypRef Expression
1 sspwb 4201 . 2 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 dfss2 3136 . 2 (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
3 vex 2733 . . . . 5 𝑥 ∈ V
43elpw 3572 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53elpw 3572 . . . 4 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
64, 5imbi12i 238 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
76albii 1463 . 2 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
81, 2, 73bitri 205 1 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1346  wcel 2141  wss 3121  𝒫 cpw 3566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589
This theorem is referenced by:  ssext  4206  nssssr  4207
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