Theorem disjsn 2437

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 2280	. . . 4 |- -. B e. (/)
2		eleq2 1532	. . . 4 |- ((A i^i {B}) = (/) -> (B e. (A i^i {B}) <-> B e. (/)))
3	1, 2	mtbiri 716	. . 3 |- ((A i^i {B}) = (/) -> -. B e. (A i^i {B}))
4		snidg 2429	. . . . 5 |- (B e. A -> B e. {B})
5	4	ancli 296	. . . 4 |- (B e. A -> (B e. A /\ B e. {B}))
6		elin 2203	. . . 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 1531	. . . . . . . 8 |- (x = B -> (x e. A <-> B e. A))
10	9	biimpcd 155	. . . . . . 7 |- (x e. A -> (x = B -> B e. A))
11		elsn 2417	. . . . . . 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 1284	. . 3 |- (-. B e. A -> A.x(x e. A -> -. x e. {B}))
16		disj1 2308	. . 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 952   = wceq 954   e. wcel 956   i^i cin 2042  (/)c0 2276  {csn 2405

This theorem is referenced by:  disjsn2 2438  orddisj 2980  ndmima 3426  limensuci 4492  php 4499  pm54.43 4552  infensuc 4618  kmlem2 4746  renfdisj 5520  cnconst 7730  sncld 7737

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-nul 2277  df-sn 2408  df-pr 2409

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