Theorem disjne 3468

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 3463	. . 3 |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B )
2		eleq1 2233	. . . . . 6 |- ( x = C -> ( x e. B <-> C e. B ) )
3	2	notbid 662	. . . . 5 |- ( x = C -> ( -. x e. B <-> -. C e. B ) )
4	3	rspccva 2833	. . . 4 |- ( ( A. x e. A -. x e. B /\ C e. A ) -> -. C e. B )
5		eleq1a 2242	. . . . 5 |- ( D e. B -> ( C = D -> C e. B ) )
6	5	necon3bd 2383	. . . 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 1195	1 |- ( ( ( A i^i B ) = (/) /\ C e. A /\ D e. B ) -> C =/= D )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   A.wral 2448   i^i cin 3120   (/)c0 3414

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-nul 3415

This theorem is referenced by: (None)