Theorem disjsn2 3091

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 3066	. . . 4 |- (B e. {A} -> B = A)
2	1	eqcomd 1889	. . 3 |- (B e. {A} -> A = B)
3	2	necon3ai 2043	. 2 |- (A =/= B -> -. B e. {A})
4		disjsn 3089	. 2 |- (({A} i^i {B}) = (/) <-> -. B e. {A})
5	3, 4	sylibr 217	1 |- (A =/= B -> ({A} i^i {B}) = (/))

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   e. wcel 1300   =/= wne 2017   i^i cin 2592  (/)c0 2875  {csn 3044

This theorem is referenced by:  xpsndisj 4339  funprg 4466  funtp 4468  phplem1 5602  pm54.43 5662  unpde2eg2 14406  repfuntw 14502  dtt2 14951

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-in 2603  df-nul 2876  df-sn 3049

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