Theorem difn0 3943

Description: If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.)

Assertion

Ref	Expression
difn0	|- ( ( A \ B ) =/= (/) -> A =/= B )

Proof of Theorem difn0

Step	Hyp	Ref	Expression
1		eqimss 3657	. . 3 |- ( A = B -> A C_ B )
2		ssdif0 3942	. . 3 |- ( A C_ B <-> ( A \ B ) = (/) )
3	1, 2	sylib 208	. 2 |- ( A = B -> ( A \ B ) = (/) )
4	3	necon3i 2826	1 |- ( ( A \ B ) =/= (/) -> A =/= B )

Syntax hints:   -> wi 4   = wceq 1483   =/= wne 2794   \ cdif 3571   C_ wss 3574   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  disjdsct  29480  bj-2upln1upl  33012