Theorem pwuncl 7805

Description: Power classes are closed under union. (Contributed by AV, 27-Feb-2024.)

Assertion

Ref	Expression
pwuncl	⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ 𝒫 𝑋)

Proof of Theorem pwuncl

Step	Hyp	Ref	Expression
1		unexg 7781	. 2 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ V)
2		elpwi 4567	. . 3 ⊢ (𝐴 ∈ 𝒫 𝑋 → 𝐴 ⊆ 𝑋)
3		elpwi 4567	. . 3 ⊢ (𝐵 ∈ 𝒫 𝑋 → 𝐵 ⊆ 𝑋)
4		unss 4191	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ↔ (𝐴 ∪ 𝐵) ⊆ 𝑋)
5	4	biimpi 219	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
6	2, 3, 5	syl2an 608	. 2 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
7	1, 6	elpwd 4566	1 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ 𝒫 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  𝒫 cpw 4560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-pw 4562  df-sn 4589  df-pr 4591  df-uni 4871

This theorem is used by:  naddunif  8728  fiin  9438  fpwipodrs  18740  pwmnd  19169  cutlt  28345  clsk1indlem3  45233  isotone1  45238  isgrtri  49244