Theorem pwunssOLD 5484

Description: Obsolete version of pwunss 4553 as of 30-Dec-2023. (Contributed by NM, 23-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
pwunssOLD	⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Proof of Theorem pwunssOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssun 4123	. . 3 ⊢ ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐴 ∪ 𝐵))
2		elun 4083	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ∈ 𝒫 𝐴 ∨ 𝑥 ∈ 𝒫 𝐵))
3		velpw 4538	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4		velpw 4538	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
5	3, 4	orbi12i 912	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∨ 𝑥 ∈ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵))
6	2, 5	bitri 274	. . 3 ⊢ (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵))
7		velpw 4538	. . 3 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∪ 𝐵))
8	1, 6, 7	3imtr4i 292	. 2 ⊢ (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) → 𝑥 ∈ 𝒫 (𝐴 ∪ 𝐵))
9	8	ssriv 3925	1 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 844   ∈ wcel 2106   ∪ cun 3885   ⊆ wss 3887  𝒫 cpw 4533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-pw 4535

This theorem is referenced by: (None)