Theorem disjne 3545

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 3540	. . 3 |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B )
2		eleq1 2292	. . . . . 6 |- ( x = C -> ( x e. B <-> C e. B ) )
3	2	notbid 671	. . . . 5 |- ( x = C -> ( -. x e. B <-> -. C e. B ) )
4	3	rspccva 2906	. . . 4 |- ( ( A. x e. A -. x e. B /\ C e. A ) -> -. C e. B )
5		eleq1a 2301	. . . . 5 |- ( D e. B -> ( C = D -> C e. B ) )
6	5	necon3bd 2443	. . . 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 284	. 2 |- ( ( ( A i^i B ) = (/) /\ C e. A ) -> ( D e. B -> C =/= D ) )
9	8	3impia 1224	1 |- ( ( ( A i^i B ) = (/) /\ C e. A /\ D e. B ) -> C =/= D )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   i^i cin 3196   (/)c0 3491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-v 2801  df-dif 3199  df-in 3203  df-nul 3492

This theorem is referenced by: (None)