Theorem disjsn 2453

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 2295	. . . 4 |- -. B e. (/)
2		eleq2 1542	. . . 4 |- ((A i^i {B}) = (/) -> (B e. (A i^i {B}) <-> B e. (/)))
3	1, 2	mtbiri 721	. . 3 |- ((A i^i {B}) = (/) -> -. B e. (A i^i {B}))
4		snidg 2445	. . . . 5 |- (B e. A -> B e. {B})
5	4	ancli 296	. . . 4 |- (B e. A -> (B e. A /\ B e. {B}))
6		elin 2218	. . . 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 1541	. . . . . . . 8 |- (x = B -> (x e. A <-> B e. A))
10	9	biimpcd 155	. . . . . . 7 |- (x e. A -> (x = B -> B e. A))
11		elsn 2433	. . . . . . 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 1290	. . 3 |- (-. B e. A -> A.x(x e. A -> -. x e. {B}))
16		disj1 2324	. . 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	impbii 157	1 |- ((A i^i {B}) = (/) <-> -. B e. A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 958   = wceq 960   e. wcel 962   i^i cin 2057  (/)c0 2291  {csn 2421

This theorem is referenced by:  disjsn2 2454  orddisj 3001  ndmima 3450  limensuci 4526  php 4533  pm54.43 4587  infensuc 4655  kmlem2 4783  unsnen 4853  renfdisj 5559  cnconst 7806  sncld 7813

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-ral 1656  df-v 1819  df-dif 2060  df-un 2061  df-in 2062  df-nul 2292  df-sn 2424  df-pr 2425

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