Theorem disjne 3273

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 3268	. . 3 |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B )
2		eleq1 2100	. . . . . 6 |- ( x = C -> ( x e. B <-> C e. B ) )
3	2	notbid 592	. . . . 5 |- ( x = C -> ( -. x e. B <-> -. C e. B ) )
4	3	rspccva 2655	. . . 4 |- ( ( A. x e. A -. x e. B /\ C e. A ) -> -. C e. B )
5		eleq1a 2109	. . . . 5 |- ( D e. B -> ( C = D -> C e. B ) )
6	5	necon3bd 2248	. . . 4 |- ( D e. B -> ( -. C e. B -> C =/= D ) )
7	4, 6	syl5com 26	. . 3 |- ( ( A. x e. A -. x e. B /\ C e. A ) -> ( D e. B -> C =/= D ) )
8	1, 7	sylanb 268	. 2 |- ( ( ( A i^i B ) = (/) /\ C e. A ) -> ( D e. B -> C =/= D ) )
9	8	3impia 1101	1 |- ( ( ( A i^i B ) = (/) /\ C e. A /\ D e. B ) -> C =/= D )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393   =/= wne 2204   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-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-nul 3225

This theorem is referenced by: (None)