Theorem pwuncl 7719

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 7692	. 2 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ V)
2		elpwi 4549	. . 3 ⊢ (𝐴 ∈ 𝒫 𝑋 → 𝐴 ⊆ 𝑋)
3		elpwi 4549	. . 3 ⊢ (𝐵 ∈ 𝒫 𝑋 → 𝐵 ⊆ 𝑋)
4		unss 4131	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) ↔ (𝐴 ∪ 𝐵) ⊆ 𝑋)
5	4	biimpi 216	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
6	2, 3, 5	syl2an 597	. 2 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
7	1, 6	elpwd 4548	1 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ 𝒫 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3430   ∪ cun 3888   ⊆ wss 3890  𝒫 cpw 4542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-un 3895  df-ss 3907  df-pw 4544  df-sn 4569  df-pr 4571  df-uni 4852

This theorem is referenced by:  naddunif  8624  fiin  9330  fpwipodrs  18501  pwmnd  18903  cutlt  27942  clsk1indlem3  44492  isotone1  44497  isgrtri  48435