Theorem pw1un 4163

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 3443	. . 3 ⊢ (∃y ∈ (A ∪ B)x = {y} ↔ (∃y ∈ A x = {y} ∨ ∃y ∈ B x = {y}))
2		elpw1 4144	. . 3 ⊢ (x ∈ ℘1(A ∪ B) ↔ ∃y ∈ (A ∪ B)x = {y})
3		elun 3220	. . . 4 ⊢ (x ∈ (℘1A ∪ ℘1B) ↔ (x ∈ ℘1A ∨ x ∈ ℘1B))
4		elpw1 4144	. . . . 5 ⊢ (x ∈ ℘1A ↔ ∃y ∈ A x = {y})
5		elpw1 4144	. . . . 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 2615   ∪ cun 3207  {csn 3737  ℘₁cpw1 4135

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 4078  ax-sn 4087

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 2478  df-ne 2518  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-1c 4136  df-pw1 4137

This theorem is referenced by:  pw1equn  4331  pw1eqadj  4332  ncfinraise  4481  tfindi  4496  tfinsuc  4498  sfindbl  4530  tcdi  6164  ce0addcnnul  6179  ce2  6192