Theorem disjsn2 2439

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 2429	. . . 4 |- (B e. {A} -> B = A)
2	1	eqcomd 1478	. . 3 |- (B e. {A} -> A = B)
3	2	necon3ai 1604	. 2 |- (A =/= B -> -. B e. {A})
4		disjsn 2438	. 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 955   e. wcel 957   =/= wne 1583   i^i cin 2043  (/)c0 2277  {csn 2406

This theorem is referenced by:  xpsndisj 3466  phplem1 4497  pm54.43 4555  dtt2 10534

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-nul 2278  df-sn 2409  df-pr 2410

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