Theorem pw1disj 4167

Description: Two unit power classes are disjoint iff the classes themselves are disjoint. (Contributed by SF, 26-Jan-2015.)

Assertion

Ref	Expression
pw1disj	|- ((~P1 A i^i ~P1 B) = (/) <-> (A i^i B) = (/))

Proof of Theorem pw1disj

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

Step	Hyp	Ref	Expression
1		disj 3591	. . . . . 6 |- ((~P1 A i^i ~P1 B) = (/) <-> A.y e. ~P1 A -. y e. ~P1 B)
2		eleq1 2413	. . . . . . . 8 |- (y = {x} -> (y e. ~P1 B <-> {x} e. ~P1 B))
3	2	notbid 285	. . . . . . 7 |- (y = {x} -> (-. y e. ~P1 B <-> -. {x} e. ~P1 B))
4	3	rspccv 2952	. . . . . 6 |- (A.y e. ~P1 A -. y e. ~P1 B -> ({x} e. ~P1 A -> -. {x} e. ~P1 B))
5	1, 4	sylbi 187	. . . . 5 |- ((~P1 A i^i ~P1 B) = (/) -> ({x} e. ~P1 A -> -. {x} e. ~P1 B))
6		snelpw1 4146	. . . . 5 |- ({x} e. ~P1 A <-> x e. A)
7		snelpw1 4146	. . . . . 6 |- ({x} e. ~P1 B <-> x e. B)
8	7	notbii 287	. . . . 5 |- (-. {x} e. ~P1 B <-> -. x e. B)
9	5, 6, 8	3imtr3g 260	. . . 4 |- ((~P1 A i^i ~P1 B) = (/) -> (x e. A -> -. x e. B))
10	9	ralrimiv 2696	. . 3 |- ((~P1 A i^i ~P1 B) = (/) -> A.x e. A -. x e. B)
11		disj 3591	. . 3 |- ((A i^i B) = (/) <-> A.x e. A -. x e. B)
12	10, 11	sylibr 203	. 2 |- ((~P1 A i^i ~P1 B) = (/) -> (A i^i B) = (/))
13		elpw1 4144	. . . . 5 |- (x e. ~P1 A <-> E.y e. A x = {y})
14		disj 3591	. . . . . . . . 9 |- ((A i^i B) = (/) <-> A.y e. A -. y e. B)
15		rsp 2674	. . . . . . . . 9 |- (A.y e. A -. y e. B -> (y e. A -> -. y e. B))
16	14, 15	sylbi 187	. . . . . . . 8 |- ((A i^i B) = (/) -> (y e. A -> -. y e. B))
17	16	imp 418	. . . . . . 7 |- (((A i^i B) = (/) /\ y e. A) -> -. y e. B)
18		eleq1 2413	. . . . . . . . 9 |- (x = {y} -> (x e. ~P1 B <-> {y} e. ~P1 B))
19		snelpw1 4146	. . . . . . . . 9 |- ({y} e. ~P1 B <-> y e. B)
20	18, 19	syl6bb 252	. . . . . . . 8 |- (x = {y} -> (x e. ~P1 B <-> y e. B))
21	20	notbid 285	. . . . . . 7 |- (x = {y} -> (-. x e. ~P1 B <-> -. y e. B))
22	17, 21	syl5ibrcom 213	. . . . . 6 |- (((A i^i B) = (/) /\ y e. A) -> (x = {y} -> -. x e. ~P1 B))
23	22	rexlimdva 2738	. . . . 5 |- ((A i^i B) = (/) -> (E.y e. A x = {y} -> -. x e. ~P1 B))
24	13, 23	syl5bi 208	. . . 4 |- ((A i^i B) = (/) -> (x e. ~P1 A -> -. x e. ~P1 B))
25	24	ralrimiv 2696	. . 3 |- ((A i^i B) = (/) -> A.x e. ~P1 A -. x e. ~P1 B)
26		disj 3591	. . 3 |- ((~P1 A i^i ~P1 B) = (/) <-> A.x e. ~P1 A -. x e. ~P1 B)
27	25, 26	sylibr 203	. 2 |- ((A i^i B) = (/) -> (~P1 A i^i ~P1 B) = (/))
28	12, 27	impbii 180	1 |- ((~P1 A i^i ~P1 B) = (/) <-> (A i^i B) = (/))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2614  E.wrex 2615   i^i cin 3208  (/)c0 3550  {csn 3737  ~P₁ cpw1 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-ral 2619  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: (None)