Theorem disjsn2 2446

Description: Intersection of distinct singletons is disjoint.

Assertion

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

Proof of Theorem disjsn2

Step	Hyp	Ref	Expression
1		elsni 2436	. . . 4 |- (B e. {A} -> B = A)
2	1	eqcomd 1483	. . 3 |- (B e. {A} -> A = B)
3	2	necon3ai 1609	. 2 |- (A =/= B -> -. B e. {A})
4		disjsn 2445	. 2 |- (({A} i^i {B}) = (/) <-> -. B e. {A})
5	3, 4	sylibr 200	1 |- (A =/= B -> ({A} i^i {B}) = (/))

Syntax hints:  -. wn 2   -> wi 3   = wceq 958   e. wcel 960   =/= wne 1588   i^i cin 2049  (/)c0 2283  {csn 2413

This theorem is referenced by:  xpsndisj 3476  phplem1 4514  pm54.43 4581  unpde2eg2 10530  dtt2 10589

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-nul 2284  df-sn 2416  df-pr 2417

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