Theorem disj 3463

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 3127	. . . 4 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2	1	eqeq1i 2178	. . 3 |- ( ( A i^i B ) = (/) <-> { x | ( x e. A /\ x e. B ) } = (/) )
3		abeq1 2280	. . 3 |- ( { x | ( x e. A /\ x e. B ) } = (/) <-> A. x ( ( x e. A /\ x e. B ) <-> x e. (/) ) )
4		imnan 685	. . . . 5 |- ( ( x e. A -> -. x e. B ) <-> -. ( x e. A /\ x e. B ) )
5		noel 3418	. . . . . 6 |- -. x e. (/)
6	5	nbn 694	. . . . 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 1463	. . 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 2453	. 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 1346   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   i^i cin 3120   (/)c0 3414

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-nul 3415

This theorem is referenced by:  disjr  3464  disj1  3465  disjne  3468  f0rn0  5392  renfdisj  7979  fvinim0ffz  10197  fxnn0nninf  10394  fprodsplitdc  11559  exmidunben  12381  dedekindeulemuub  13389  dedekindeulemlu  13393  dedekindicclemuub  13398  dedekindicclemlu  13402  ivthinclemdisj  13412  exmidsbthrlem  14054