Theorem bj-sselpwuni 37714

Description: Quantitative version of ssexg 5289: 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 5289	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
2		ssuni 4897	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ⊆ ∪ 𝑉)
3	1, 2	elpwd 4567	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝑉)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4561  ∪ cuni 4871

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-in 3911  df-ss 3921  df-pw 4563  df-uni 4872

This theorem is used by:  bj-unirel  37715