Theorem disj 3416

Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)

Assertion

Ref	Expression
disj	|- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B )

Distinct variable groups:   x, A   x, B

Proof of Theorem disj

Step	Hyp	Ref	Expression
1		df-in 3082	. . . 4 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2	1	eqeq1i 2148	. . 3 |- ( ( A i^i B ) = (/) <-> { x | ( x e. A /\ x e. B ) } = (/) )
3		abeq1 2250	. . 3 |- ( { x | ( x e. A /\ x e. B ) } = (/) <-> A. x ( ( x e. A /\ x e. B ) <-> x e. (/) ) )
4		imnan 680	. . . . 5 |- ( ( x e. A -> -. x e. B ) <-> -. ( x e. A /\ x e. B ) )
5		noel 3372	. . . . . 6 |- -. x e. (/)
6	5	nbn 689	. . . . 5 |- ( -. ( x e. A /\ x e. B ) <-> ( ( x e. A /\ x e. B ) <-> x e. (/) ) )
7	4, 6	bitr2i 184	. . . 4 |- ( ( ( x e. A /\ x e. B ) <-> x e. (/) ) <-> ( x e. A -> -. x e. B ) )
8	7	albii 1447	. . 3 |- ( A. x ( ( x e. A /\ x e. B ) <-> x e. (/) ) <-> A. x ( x e. A -> -. x e. B ) )
9	2, 3, 8	3bitri 205	. 2 |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) )
10		df-ral 2422	. 2 |- ( A. x e. A -. x e. B <-> A. x ( x e. A -> -. x e. B ) )
11	9, 10	bitr4i 186	1 |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1330   = wceq 1332   e. wcel 1481   {cab 2126   A.wral 2417   i^i cin 3075   (/)c0 3368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-dif 3078  df-in 3082  df-nul 3369

This theorem is referenced by:  disjr  3417  disj1  3418  disjne  3421  f0rn0  5325  renfdisj  7848  fvinim0ffz  10049  fxnn0nninf  10242  exmidunben  11975  dedekindeulemuub  12803  dedekindeulemlu  12807  dedekindicclemuub  12812  dedekindicclemlu  12816  ivthinclemdisj  12826  exmidsbthrlem  13392