Theorem disjsn2 3788

Description: Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)

Assertion

Ref	Expression
disjsn2	|- (A =/= B -> ({A} i^i {B}) = (/))

Proof of Theorem disjsn2

Step	Hyp	Ref	Expression
1		elsni 3758	. . . 4 |- (B e. {A} -> B = A)
2	1	eqcomd 2358	. . 3 |- (B e. {A} -> A = B)
3	2	necon3ai 2557	. 2 |- (A =/= B -> -. B e. {A})
4		disjsn 3787	. 2 |- (({A} i^i {B}) = (/) <-> -. B e. {A})
5	3, 4	sylibr 203	1 |- (A =/= B -> ({A} i^i {B}) = (/))

Syntax hints:  -. wn 3   -> wi 4   = wceq 1642   e. wcel 1710   =/= wne 2517   i^i cin 3209  (/)c0 3551  {csn 3738

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-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  df-sn 3742

This theorem is referenced by:  difprsn1  3848  diftpsn3  3850  xpsndisj  5050  funprg  5150  funprgOLD  5151