Theorem bj-sselpwuni 37033

Description: Quantitative version of ssexg 5329: 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 5329	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
2		ssuni 4937	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ⊆ ∪ 𝑉)
3	1, 2	elpwd 4611	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝑉)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607  df-uni 4913

This theorem is referenced by:  bj-unirel  37034