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Theorem bj-elpwg 34332
Description: If the intersection of two classes is a set, then inclusion among these classes is equivalent to membership in the powerclass. Common generalization of elpwg 4543 and elpw2g 5238 (the latter of which could be proved from it). (Contributed by BJ, 31-Dec-2023.)
Assertion
Ref Expression
bj-elpwg ((𝐴𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Proof of Theorem bj-elpwg
StepHypRef Expression
1 elpwi 4549 . 2 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
2 ssidd 3988 . . . . . 6 (𝐴𝐵𝐴𝐴)
3 id 22 . . . . . 6 (𝐴𝐵𝐴𝐵)
42, 3ssind 4207 . . . . 5 (𝐴𝐵𝐴 ⊆ (𝐴𝐵))
5 ssexg 5218 . . . . 5 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ 𝑉) → 𝐴 ∈ V)
64, 5sylan 582 . . . 4 ((𝐴𝐵 ∧ (𝐴𝐵) ∈ 𝑉) → 𝐴 ∈ V)
7 elpwg 4543 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
87biimparc 482 . . . 4 ((𝐴𝐵𝐴 ∈ V) → 𝐴 ∈ 𝒫 𝐵)
96, 8syldan 593 . . 3 ((𝐴𝐵 ∧ (𝐴𝐵) ∈ 𝑉) → 𝐴 ∈ 𝒫 𝐵)
109expcom 416 . 2 ((𝐴𝐵) ∈ 𝑉 → (𝐴𝐵𝐴 ∈ 𝒫 𝐵))
111, 10impbid2 228 1 ((𝐴𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wcel 2107  Vcvv 3493  cin 3933  wss 3934  𝒫 cpw 4537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-in 3941  df-ss 3950  df-pw 4539
This theorem is referenced by: (None)
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