Theorem disjne 3597

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 3592	. . 3 |- ((A i^i B) = (/) <-> A.x e. A -. x e. B)
2		eleq1 2413	. . . . . 6 |- (x = C -> (x e. B <-> C e. B))
3	2	notbid 285	. . . . 5 |- (x = C -> (-. x e. B <-> -. C e. B))
4	3	rspccva 2955	. . . 4 |- ((A.x e. A -. x e. B /\ C e. A) -> -. C e. B)
5		eleq1a 2422	. . . . 5 |- (D e. B -> (C = D -> C e. B))
6	5	necon3bd 2554	. . . 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 458	. 2 |- (((A i^i B) = (/) /\ C e. A) -> (D e. B -> C =/= D))
9	8	3impia 1148	1 |- (((A i^i B) = (/) /\ C e. A /\ D e. B) -> C =/= D)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710   =/= wne 2517  A.wral 2615   i^i cin 3209  (/)c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)