Theorem intunsn 2565

Description: Theorem joining a singleton to an intersection.

Hypothesis

Ref	Expression
intunsn.1	|- B e. V

Assertion

Ref	Expression
intunsn	|- |^|(A u. {B}) = (|^|A i^i B)

Proof of Theorem intunsn

Step	Hyp	Ref	Expression
1		intun 2562	. 2 |- |^|(A u. {B}) = (|^|A i^i |^|{B})
2		intunsn.1	. . . 4 |- B e. V
3	2	intsn 2564	. . 3 |- |^|{B} = B
4	3	ineq2i 2214	. 2 |- (|^|A i^i |^|{B}) = (|^|A i^i B)
5	1, 4	eqtr 1495	1 |- |^|(A u. {B}) = (|^|A i^i B)

Syntax hints:   = wceq 956   e. wcel 958  Vcvv 1811   u. cun 2045   i^i cin 2046  {csn 2409  |^|cint 2533

This theorem is referenced by:  fiint 4559  fiintOLD 4560

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-un 2050  df-in 2051  df-sn 2412  df-pr 2413  df-int 2534

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