Theorem disjne 3416

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 3411	. . 3 |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B )
2		eleq1 2202	. . . . . 6 |- ( x = C -> ( x e. B <-> C e. B ) )
3	2	notbid 656	. . . . 5 |- ( x = C -> ( -. x e. B <-> -. C e. B ) )
4	3	rspccva 2788	. . . 4 |- ( ( A. x e. A -. x e. B /\ C e. A ) -> -. C e. B )
5		eleq1a 2211	. . . . 5 |- ( D e. B -> ( C = D -> C e. B ) )
6	5	necon3bd 2351	. . . 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 1178	1 |- ( ( ( A i^i B ) = (/) /\ C e. A /\ D e. B ) -> C =/= D )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   =/= wne 2308   A.wral 2416   i^i cin 3070   (/)c0 3363

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-v 2688  df-dif 3073  df-in 3077  df-nul 3364

This theorem is referenced by: (None)