Theorem disjne 3267

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 3262	. . 3 |- ((A i^i B) = (/) <-> A.x e. A -. x e. B)
2		eleq1 2097	. . . . . 6 |- (x = C -> (x e. B <-> C e. B))
3	2	notbid 591	. . . . 5 |- (x = C -> (-. x e. B <-> -. C e. B))
4	3	rspccva 2649	. . . 4 |- ((A.x e. A -. x e. B /\ C e. A) -> -. C e. B)
5		eleq1a 2106	. . . . 5 |- ( D e. B -> ( C = D -> C e. B))
6	5	necon3bd 2242	. . . 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 1100	1 |- (((A i^i B) = (/) /\ C e. A /\ D e. B) -> C =/= D)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   /\ w3a 884   = wceq 1242   e. wcel 1390   =/= wne 2201  A.wral 2300   i^i cin 2910   (/)c0 3218

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-nul 3219

This theorem is referenced by: (None)