Theorem disj 3268

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 2924	. . . 4 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
2	1	eqeq1i 2047	. . 3 |- ( ( A i^i B ) = (/) <-> { x | ( x e. A /\ x e. B ) } = (/) )
3		abeq1 2147	. . 3 |- ( { x | ( x e. A /\ x e. B ) } = (/) <-> A. x ( ( x e. A /\ x e. B ) <-> x e. (/) ) )
4		imnan 624	. . . . 5 |- ( ( x e. A -> -. x e. B ) <-> -. ( x e. A /\ x e. B ) )
5		noel 3228	. . . . . 6 |- -. x e. (/)
6	5	nbn 615	. . . . 5 |- ( -. ( x e. A /\ x e. B ) <-> ( ( x e. A /\ x e. B ) <-> x e. (/) ) )
7	4, 6	bitr2i 174	. . . 4 |- ( ( ( x e. A /\ x e. B ) <-> x e. (/) ) <-> ( x e. A -> -. x e. B ) )
8	7	albii 1359	. . 3 |- ( A. x ( ( x e. A /\ x e. B ) <-> x e. (/) ) <-> A. x ( x e. A -> -. x e. B ) )
9	2, 3, 8	3bitri 195	. 2 |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) )
10		df-ral 2311	. 2 |- ( A. x e. A -. x e. B <-> A. x ( x e. A -> -. x e. B ) )
11	9, 10	bitr4i 176	1 |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026   A.wral 2306   i^i cin 2916   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-nul 3225

This theorem is referenced by:  disjr  3269  disj1  3270  disjne  3273  renfdisj  7079  fvinim0ffz  9096