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Theorem ssextss 3947
 Description: An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
Assertion
Ref Expression
ssextss (ABx(xAxB))
Distinct variable groups:   x,A   x,B

Proof of Theorem ssextss
StepHypRef Expression
1 sspwb 3943 . 2 (AB ↔ 𝒫 A ⊆ 𝒫 B)
2 dfss2 2928 . 2 (𝒫 A ⊆ 𝒫 Bx(x 𝒫 Ax 𝒫 B))
3 vex 2554 . . . . 5 x V
43elpw 3357 . . . 4 (x 𝒫 AxA)
53elpw 3357 . . . 4 (x 𝒫 BxB)
64, 5imbi12i 228 . . 3 ((x 𝒫 Ax 𝒫 B) ↔ (xAxB))
76albii 1356 . 2 (x(x 𝒫 Ax 𝒫 B) ↔ x(xAxB))
81, 2, 73bitri 195 1 (ABx(xAxB))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   ∈ wcel 1390   ⊆ wss 2911  𝒫 cpw 3351 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-bnd 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:  ssext  3948  nssssr  3949
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