Theorem pw1un 4164

Description: Unit power class distributes over union. (Contributed by SF, 22-Jan-2015.)

Assertion

Ref	Expression
pw1un	⊢ ℘1(A ∪ B) = (℘1A ∪ ℘1B)

Proof of Theorem pw1un

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

Step	Hyp	Ref	Expression
1		rexun 3444	. . 3 ⊢ (∃y ∈ (A ∪ B)x = {y} ↔ (∃y ∈ A x = {y} ∨ ∃y ∈ B x = {y}))
2		elpw1 4145	. . 3 ⊢ (x ∈ ℘1(A ∪ B) ↔ ∃y ∈ (A ∪ B)x = {y})
3		elun 3221	. . . 4 ⊢ (x ∈ (℘1A ∪ ℘1B) ↔ (x ∈ ℘1A ∨ x ∈ ℘1B))
4		elpw1 4145	. . . . 5 ⊢ (x ∈ ℘1A ↔ ∃y ∈ A x = {y})
5		elpw1 4145	. . . . 5 ⊢ (x ∈ ℘1B ↔ ∃y ∈ B x = {y})
6	4, 5	orbi12i 507	. . . 4 ⊢ ((x ∈ ℘1A ∨ x ∈ ℘1B) ↔ (∃y ∈ A x = {y} ∨ ∃y ∈ B x = {y}))
7	3, 6	bitri 240	. . 3 ⊢ (x ∈ (℘1A ∪ ℘1B) ↔ (∃y ∈ A x = {y} ∨ ∃y ∈ B x = {y}))
8	1, 2, 7	3bitr4i 268	. 2 ⊢ (x ∈ ℘1(A ∪ B) ↔ x ∈ (℘1A ∪ ℘1B))
9	8	eqriv 2350	1 ⊢ ℘1(A ∪ B) = (℘1A ∪ ℘1B)

Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ∪ cun 3208  {csn 3738  ℘₁cpw1 4136

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-rex 2621  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-1c 4137  df-pw1 4138

This theorem is referenced by:  pw1equn  4332  pw1eqadj  4333  ncfinraise  4482  tfindi  4497  tfinsuc  4499  sfindbl  4531  tcdi  6165  ce0addcnnul  6180  ce2  6193