Theorem disjel 3873

Description: A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)

Assertion

Ref	Expression
disjel

Proof of Theorem disjel

Step	Hyp	Ref	Expression
1		disj3 3871	. . 3
2		eleq2 2530	. . . 4
3		eldifn 3626	. . . 4
4	2, 3	syl6bi 228	. . 3
5	1, 4	sylbi 195	. 2
6	5	imp 429	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  \cdif 3472  i^icin 3474  c0 3784

This theorem is referenced by:  disjxun  4450  fvun1  5944  dedekindle  9766  fprodsplit  13770  dvasin  30103

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-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-dif 3478  df-in 3482  df-nul 3785