Theorem bj-sselpwuni 37034

Description: Quantitative version of ssexg 5262: a subset of an element of a class is an element of the powerclass of the union of that class. (Contributed by BJ, 6-Apr-2024.)

Assertion

Ref	Expression
bj-sselpwuni	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝑉)

Proof of Theorem bj-sselpwuni

Step	Hyp	Ref	Expression
1		ssexg 5262	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
2		ssuni 4883	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ⊆ ∪ 𝑉)
3	1, 2	elpwd 4557	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝑉)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3436   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-in 3910  df-ss 3920  df-pw 4553  df-uni 4859

This theorem is referenced by:  bj-unirel  37035