Theorem pwunss 4119

Description: The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)

Assertion

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

Proof of Theorem pwunss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssun 3180	. . 3 ⊢ ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐴 ∪ 𝐵))
2		elun 3142	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ∈ 𝒫 𝐴 ∨ 𝑥 ∈ 𝒫 𝐵))
3		vex 2623	. . . . . 6 ⊢ 𝑥 ∈ V
4	3	elpw 3439	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
5	3	elpw 3439	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
6	4, 5	orbi12i 717	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∨ 𝑥 ∈ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵))
7	2, 6	bitri 183	. . 3 ⊢ (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐵))
8	3	elpw 3439	. . 3 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∪ 𝐵))
9	1, 7, 8	3imtr4i 200	. 2 ⊢ (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) → 𝑥 ∈ 𝒫 (𝐴 ∪ 𝐵))
10	9	ssriv 3030	1 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 665   ∈ wcel 1439   ∪ cun 2998   ⊆ wss 3000  𝒫 cpw 3433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435

This theorem is referenced by:  pwundifss  4121  pwunim  4122