Theorem unipw 4118

Description: A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (The proof was shortened by Alan Sare, 28-Dec-2008.) (Contributed by SF, 14-Oct-1996.) (Revised by SF, 29-Dec-2008.)

Assertion

Ref	Expression
unipw	⊢ ∪℘A = A

Proof of Theorem unipw

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 3895	. . . 4 ⊢ (x ∈ ∪℘A ↔ ∃y(x ∈ y ∧ y ∈ ℘A))
2		vex 2863	. . . . . . . 8 ⊢ y ∈ V
3	2	elpw 3729	. . . . . . 7 ⊢ (y ∈ ℘A ↔ y ⊆ A)
4		ssel 3268	. . . . . . 7 ⊢ (y ⊆ A → (x ∈ y → x ∈ A))
5	3, 4	sylbi 187	. . . . . 6 ⊢ (y ∈ ℘A → (x ∈ y → x ∈ A))
6	5	impcom 419	. . . . 5 ⊢ ((x ∈ y ∧ y ∈ ℘A) → x ∈ A)
7	6	exlimiv 1634	. . . 4 ⊢ (∃y(x ∈ y ∧ y ∈ ℘A) → x ∈ A)
8	1, 7	sylbi 187	. . 3 ⊢ (x ∈ ∪℘A → x ∈ A)
9		vex 2863	. . . . 5 ⊢ x ∈ V
10	9	snid 3761	. . . 4 ⊢ x ∈ {x}
11		snelpwi 4117	. . . 4 ⊢ (x ∈ A → {x} ∈ ℘A)
12		elunii 3897	. . . 4 ⊢ ((x ∈ {x} ∧ {x} ∈ ℘A) → x ∈ ∪℘A)
13	10, 11, 12	sylancr 644	. . 3 ⊢ (x ∈ A → x ∈ ∪℘A)
14	8, 13	impbii 180	. 2 ⊢ (x ∈ ∪℘A ↔ x ∈ A)
15	14	eqriv 2350	1 ⊢ ∪℘A = A

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ⊆ wss 3258  ℘cpw 3723  {csn 3738  ∪cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-uni 3893

This theorem is referenced by:  nnadjoinpw  4522