Theorem pwunss 1916

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.

Assertion

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

Proof of Theorem pwunss

Step	Hyp	Ref	Expression
1		ssun 1634	. . 3 ⊢ ((x ⊆ A ∨ x ⊆ B) → x ⊆ (A ∪ B))
2		elun 1601	. . . 4 ⊢ (x ∈ (℘A ∪ ℘B) ↔ (x ∈ ℘A ∨ x ∈ ℘B))
3		visset 1350	. . . . . 6 ⊢ x ∈ V
4	3	elpw 1801	. . . . 5 ⊢ (x ∈ ℘A ↔ x ⊆ A)
5	3	elpw 1801	. . . . 5 ⊢ (x ∈ ℘B ↔ x ⊆ B)
6	4, 5	orbi12i 216	. . . 4 ⊢ ((x ∈ ℘A ∨ x ∈ ℘B) ↔ (x ⊆ A ∨ x ⊆ B))
7	2, 6	bitr 151	. . 3 ⊢ (x ∈ (℘A ∪ ℘B) ↔ (x ⊆ A ∨ x ⊆ B))
8	3	elpw 1801	. . 3 ⊢ (x ∈ ℘(A ∪ B) ↔ x ⊆ (A ∪ B))
9	1, 7, 8	3imtr4 192	. 2 ⊢ (x ∈ (℘A ∪ ℘B) → x ∈ ℘(A ∪ B))
10	9	ssriv 1508	1 ⊢ (℘A ∪ ℘B) ⊆ ℘(A ∪ B)

Syntax hints:   ∨ wo 195   ∈ wcel 1092   ∪ cun 1485   ⊆ wss 1487  ℘cpw 1798

This theorem is referenced by:  pwun 1918

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-pw 1799

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