Theorem disjne 3462

Description: Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
disjne	|- ( ( ( A i^i B ) = (/) /\ C e. A /\ D e. B ) -> C =/= D )

Proof of Theorem disjne

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disj 3457	. . 3 |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B )
2		eleq1 2229	. . . . . 6 |- ( x = C -> ( x e. B <-> C e. B ) )
3	2	notbid 657	. . . . 5 |- ( x = C -> ( -. x e. B <-> -. C e. B ) )
4	3	rspccva 2829	. . . 4 |- ( ( A. x e. A -. x e. B /\ C e. A ) -> -. C e. B )
5		eleq1a 2238	. . . . 5 |- ( D e. B -> ( C = D -> C e. B ) )
6	5	necon3bd 2379	. . . 4 |- ( D e. B -> ( -. C e. B -> C =/= D ) )
7	4, 6	syl5com 29	. . 3 |- ( ( A. x e. A -. x e. B /\ C e. A ) -> ( D e. B -> C =/= D ) )
8	1, 7	sylanb 282	. 2 |- ( ( ( A i^i B ) = (/) /\ C e. A ) -> ( D e. B -> C =/= D ) )
9	8	3impia 1190	1 |- ( ( ( A i^i B ) = (/) /\ C e. A /\ D e. B ) -> C =/= D )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2336   A.wral 2444   i^i cin 3115   (/)c0 3409

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-v 2728  df-dif 3118  df-in 3122  df-nul 3410

This theorem is referenced by: (None)