Theorem disj 3471

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 3135	. . . 4 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2	1	eqeq1i 2185	. . 3 |- ( ( A i^i B ) = (/) <-> { x | ( x e. A /\ x e. B ) } = (/) )
3		abeq1 2287	. . 3 |- ( { x | ( x e. A /\ x e. B ) } = (/) <-> A. x ( ( x e. A /\ x e. B ) <-> x e. (/) ) )
4		imnan 690	. . . . 5 |- ( ( x e. A -> -. x e. B ) <-> -. ( x e. A /\ x e. B ) )
5		noel 3426	. . . . . 6 |- -. x e. (/)
6	5	nbn 699	. . . . 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 1470	. . 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 2460	. 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 1351   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   i^i cin 3128   (/)c0 3422

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-dif 3131  df-in 3135  df-nul 3423

This theorem is referenced by:  disjr  3472  disj1  3473  disjne  3476  f0rn0  5405  renfdisj  7994  fvinim0ffz  10214  fxnn0nninf  10411  fprodsplitdc  11575  exmidunben  12397  dedekindeulemuub  13728  dedekindeulemlu  13732  dedekindicclemuub  13737  dedekindicclemlu  13741  ivthinclemdisj  13751  exmidsbthrlem  14393