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Theorem ssextss 3983
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 3979 . 2 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 dfss2 2961 . 2 (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
3 vex 2577 . . . . 5 𝑥 ∈ V
43elpw 3392 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53elpw 3392 . . . 4 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
64, 5imbi12i 232 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
76albii 1375 . 2 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
81, 2, 73bitri 199 1 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wcel 1409  wss 2944  𝒫 cpw 3386
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:  ssext  3984  nssssr  3985
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