Theorem disjne 3872

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

Proof of Theorem disjne

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disj 3867	. . 3
2		eleq1 2529	. . . . . 6
3	2	notbid 294	. . . . 5
4	3	rspccva 3209	. . . 4
5		eleq1a 2540	. . . . 5
6	5	necon3bd 2669	. . . 4
7	4, 6	syl5com 30	. . 3
8	1, 7	sylanb 472	. 2
9	8	3impia 1193	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  i^icin 3474  c0 3784

This theorem is referenced by:  brdom7disj  8930  brdom6disj  8931  frlmssuvc1  18825  frlmssuvc1OLD  18827  frlmsslsp  18829  frlmsslspOLD  18830  kelac1  31009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-v 3111  df-dif 3478  df-in 3482  df-nul 3785