Theorem difsnid 2458

Description: If we remove a single element from a class then put it back in, we end up with the original class.

Assertion

Ref	Expression
difsnid	|- (B e. A -> ((A \ {B}) u. {B}) = A)

Proof of Theorem difsnid

Step	Hyp	Ref	Expression
1		snssi 2457	. . 3 |- (B e. A -> {B} (_ A)
2		undif 2333	. . 3 |- ({B} (_ A <-> ({B} u. (A \ {B})) = A)
3	1, 2	sylib 198	. 2 |- (B e. A -> ({B} u. (A \ {B})) = A)
4		uncom 2166	. 2 |- ((A \ {B}) u. {B}) = ({B} u. (A \ {B}))
5	3, 4	syl5eq 1511	1 |- (B e. A -> ((A \ {B}) u. {B}) = A)

Syntax hints:   -> wi 3   = wceq 953   e. wcel 955   \ cdif 2034   u. cun 2035   (_ wss 2037  {csn 2399

This theorem is referenced by:  phplem2 4489  pssnn 4513  moec 10357

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-sn 2402

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