Theorem disjsn 2431

Description: Intersection with the singleton of a non-member is disjoint.

Assertion

Ref	Expression
disjsn	|- ((A i^i {B}) = (/) <-> -. B e. A)

Proof of Theorem disjsn

Step	Hyp	Ref	Expression
1		noel 2274	. . . 4 |- -. B e. (/)
2		eleq2 1527	. . . 4 |- ((A i^i {B}) = (/) -> (B e. (A i^i {B}) <-> B e. (/)))
3	1, 2	mtbiri 715	. . 3 |- ((A i^i {B}) = (/) -> -. B e. (A i^i {B}))
4		snidg 2423	. . . . 5 |- (B e. A -> B e. {B})
5	4	ancli 296	. . . 4 |- (B e. A -> (B e. A /\ B e. {B}))
6		elin 2197	. . . 4 |- (B e. (A i^i {B}) <-> (B e. A /\ B e. {B}))
7	5, 6	sylibr 200	. . 3 |- (B e. A -> B e. (A i^i {B}))
8	3, 7	nsyl 116	. 2 |- ((A i^i {B}) = (/) -> -. B e. A)
9		eleq1 1526	. . . . . . . 8 |- (x = B -> (x e. A <-> B e. A))
10	9	biimpcd 155	. . . . . . 7 |- (x e. A -> (x = B -> B e. A))
11		elsn 2411	. . . . . . 7 |- (x e. {B} <-> x = B)
12	10, 11	syl5ib 206	. . . . . 6 |- (x e. A -> (x e. {B} -> B e. A))
13	12	con3d 95	. . . . 5 |- (x e. A -> (-. B e. A -> -. x e. {B}))
14	13	com12 11	. . . 4 |- (-. B e. A -> (x e. A -> -. x e. {B}))
15	14	19.21aiv 1281	. . 3 |- (-. B e. A -> A.x(x e. A -> -. x e. {B}))
16		disj1 2302	. . 3 |- ((A i^i {B}) = (/) <-> A.x(x e. A -> -. x e. {B}))
17	15, 16	sylibr 200	. 2 |- (-. B e. A -> (A i^i {B}) = (/))
18	8, 17	impbi 157	1 |- ((A i^i {B}) = (/) <-> -. B e. A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955   i^i cin 2036  (/)c0 2270  {csn 2399

This theorem is referenced by:  disjsn2 2432  orddisj 2975  ndmima 3418  limensuci 4486  php 4493  pm54.43 4546  infensuc 4610  kmlem2 4738  renfdisj 5512  cnconst 7719  sncld 7726

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-nul 2271  df-sn 2402  df-pr 2403

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