Theorem disjsn2 2503

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 2493	. . . 4 |- (B e. {A} -> B = A)
2	1	eqcomd 1523	. . 3 |- (B e. {A} -> A = B)
3	2	necon3ai 1649	. 2 |- (A =/= B -> -. B e. {A})
4		disjsn 2502	. 2 |- (({A} i^i {B}) = (/) <-> -. B e. {A})
5	3, 4	sylibr 198	1 |- (A =/= B -> ({A} i^i {B}) = (/))

Syntax hints:  -. wn 2   -> wi 3   = wceq 992   e. wcel 994   =/= wne 1628   i^i cin 2098  (/)c0 2332  {csn 2467

This theorem is referenced by:  xpsndisj 3555  phplem1 4655  pm54.43 4715  unpde2eg2 10825  dtt2 11110

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-nul 2333  df-sn 2470  df-pr 2471

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