Theorem disj 3509

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 3172	. . . 4 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2	1	eqeq1i 2213	. . 3 |- ( ( A i^i B ) = (/) <-> { x | ( x e. A /\ x e. B ) } = (/) )
3		abeq1 2315	. . 3 |- ( { x | ( x e. A /\ x e. B ) } = (/) <-> A. x ( ( x e. A /\ x e. B ) <-> x e. (/) ) )
4		imnan 692	. . . . 5 |- ( ( x e. A -> -. x e. B ) <-> -. ( x e. A /\ x e. B ) )
5		noel 3464	. . . . . 6 |- -. x e. (/)
6	5	nbn 701	. . . . 5 |- ( -. ( x e. A /\ x e. B ) <-> ( ( x e. A /\ x e. B ) <-> x e. (/) ) )
7	4, 6	bitr2i 185	. . . 4 |- ( ( ( x e. A /\ x e. B ) <-> x e. (/) ) <-> ( x e. A -> -. x e. B ) )
8	7	albii 1493	. . 3 |- ( A. x ( ( x e. A /\ x e. B ) <-> x e. (/) ) <-> A. x ( x e. A -> -. x e. B ) )
9	2, 3, 8	3bitri 206	. 2 |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) )
10		df-ral 2489	. 2 |- ( A. x e. A -. x e. B <-> A. x ( x e. A -> -. x e. B ) )
11	9, 10	bitr4i 187	1 |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2176   {cab 2191   A.wral 2484   i^i cin 3165   (/)c0 3460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-dif 3168  df-in 3172  df-nul 3461

This theorem is referenced by:  disjr  3510  disj1  3511  disjne  3514  f0rn0  5470  renfdisj  8132  fvinim0ffz  10370  xnn0nnen  10582  fxnn0nninf  10584  fprodsplitdc  11907  exmidunben  12797  dedekindeulemuub  15089  dedekindeulemlu  15093  dedekindicclemuub  15098  dedekindicclemlu  15102  ivthinclemdisj  15112  exmidsbthrlem  15961