Theorem pwun 1918

Description: The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28.

Assertion

Ref	Expression
pwun	⊢ ((A ⊆ B ∨ B ⊆ A) ↔ ℘(A ∪ B) = (℘A ∪ ℘B))

Proof of Theorem pwun

Step	Hyp	Ref	Expression
1		pwunss 1916	. . 3 ⊢ (℘A ∪ ℘B) ⊆ ℘(A ∪ B)
2	1	biantru 543	. 2 ⊢ (℘(A ∪ B) ⊆ (℘A ∪ ℘B) ↔ (℘(A ∪ B) ⊆ (℘A ∪ ℘B) ∧ (℘A ∪ ℘B) ⊆ ℘(A ∪ B)))
3		pwssun 1917	. 2 ⊢ ((A ⊆ B ∨ B ⊆ A) ↔ ℘(A ∪ B) ⊆ (℘A ∪ ℘B))
4		eqss 1516	. 2 ⊢ (℘(A ∪ B) = (℘A ∪ ℘B) ↔ (℘(A ∪ B) ⊆ (℘A ∪ ℘B) ∧ (℘A ∪ ℘B) ⊆ ℘(A ∪ B)))
5	2, 3, 4	3bitr4 158	1 ⊢ ((A ⊆ B ∨ B ⊆ A) ↔ ℘(A ∪ B) = (℘A ∪ ℘B))

Syntax hints:   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∪ cun 1485   ⊆ wss 1487  ℘cpw 1798

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

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-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812

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