Theorem pwuncl 7491

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 7470	. 2 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ V)
2		elpwi 4503	. . 3 ⊢ (𝐴 ∈ 𝒫 𝑋 → 𝐴 ⊆ 𝑋)
3		elpwi 4503	. . 3 ⊢ (𝐵 ∈ 𝒫 𝑋 → 𝐵 ⊆ 𝑋)
4		unss 4089	. . . 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 598	. 2 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ⊆ 𝑋)
7	1, 6	elpwd 4502	1 ⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ 𝒫 𝑋)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  Vcvv 3409   ∪ cun 3856   ⊆ wss 3858  𝒫 cpw 4494

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-pw 4496  df-sn 4523  df-pr 4525  df-uni 4799

This theorem is referenced by:  fiin  8919  fpwipodrs  17840  pwmnd  18168  madebdayim  33627  clsk1indlem3  41119  isotone1  41124