Theorem pw1in 4165

Description: Unit power class distributes over intersection. (Contributed by SF, 13-Feb-2015.)

Assertion

Ref	Expression
pw1in	|- ~P1 (A i^i B) = (~P1 A i^i ~P1 B)

Proof of Theorem pw1in

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

Step	Hyp	Ref	Expression
1		ancom 437	. . . . 5 |- (((y e. A /\ x e. ~P1 B) /\ x = {y}) <-> (x = {y} /\ (y e. A /\ x e. ~P1 B)))
2		eleq1 2413	. . . . . . . . 9 |- (x = {y} -> (x e. ~P1 B <-> {y} e. ~P1 B))
3		snelpw1 4147	. . . . . . . . 9 |- ({y} e. ~P1 B <-> y e. B)
4	2, 3	syl6bb 252	. . . . . . . 8 |- (x = {y} -> (x e. ~P1 B <-> y e. B))
5	4	anbi2d 684	. . . . . . 7 |- (x = {y} -> ((y e. A /\ x e. ~P1 B) <-> (y e. A /\ y e. B)))
6		elin 3220	. . . . . . 7 |- (y e. (A i^i B) <-> (y e. A /\ y e. B))
7	5, 6	syl6bbr 254	. . . . . 6 |- (x = {y} -> ((y e. A /\ x e. ~P1 B) <-> y e. (A i^i B)))
8	7	pm5.32ri 619	. . . . 5 |- (((y e. A /\ x e. ~P1 B) /\ x = {y}) <-> (y e. (A i^i B) /\ x = {y}))
9		an12 772	. . . . 5 |- ((x = {y} /\ (y e. A /\ x e. ~P1 B)) <-> (y e. A /\ (x = {y} /\ x e. ~P1 B)))
10	1, 8, 9	3bitr3i 266	. . . 4 |- ((y e. (A i^i B) /\ x = {y}) <-> (y e. A /\ (x = {y} /\ x e. ~P1 B)))
11	10	rexbii2 2644	. . 3 |- (E.y e. (A i^i B)x = {y} <-> E.y e. A (x = {y} /\ x e. ~P1 B))
12		elpw1 4145	. . 3 |- (x e. ~P1 (A i^i B) <-> E.y e. (A i^i B)x = {y})
13		elpw1 4145	. . . . 5 |- (x e. ~P1 A <-> E.y e. A x = {y})
14	13	anbi1i 676	. . . 4 |- ((x e. ~P1 A /\ x e. ~P1 B) <-> (E.y e. A x = {y} /\ x e. ~P1 B))
15		elin 3220	. . . 4 |- (x e. (~P1 A i^i ~P1 B) <-> (x e. ~P1 A /\ x e. ~P1 B))
16		r19.41v 2765	. . . 4 |- (E.y e. A (x = {y} /\ x e. ~P1 B) <-> (E.y e. A x = {y} /\ x e. ~P1 B))
17	14, 15, 16	3bitr4i 268	. . 3 |- (x e. (~P1 A i^i ~P1 B) <-> E.y e. A (x = {y} /\ x e. ~P1 B))
18	11, 12, 17	3bitr4i 268	. 2 |- (x e. ~P1 (A i^i B) <-> x e. (~P1 A i^i ~P1 B))
19	18	eqriv 2350	1 |- ~P1 (A i^i B) = (~P1 A i^i ~P1 B)

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2616   i^i cin 3209  {csn 3738  ~P₁ cpw1 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:  tfindi  4497  tcdi  6165  ce0addcnnul  6180