Theorem intunsn 3435

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 3430	. 2 |- |^|(A u. {B}) = (|^|A i^i |^|{B})
2		intunsn.1	. . . 4 |- B e. _V
3	2	intsn 3433	. . 3 |- |^|{B} = B
4	3	ineq2i 3006	. 2 |- (|^|A i^i |^|{B}) = (|^|A i^i B)
5	1, 4	eqtri 2161	1 |- |^|(A u. {B}) = (|^|A i^i B)

Syntax hints:   = wceq 1586   e. wcel 1588  _Vcvv 2538   u. cun 2825   i^i cin 2826  {csn 3238  |^|cint 3400

This theorem is referenced by:  fiint 5872  inficl 16581

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1592  ax-gen 1593  ax-8 1594  ax-9 1595  ax-10 1596  ax-11 1597  ax-12 1598  ax-17 1605  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-10o 1781  ax-16 1854  ax-11o 1864  ax-ext 2123

This theorem depends on definitions:  df-bi 220  df-or 338  df-an 339  df-ex 1616  df-sb 1816  df-clab 2129  df-cleq 2134  df-clel 2137  df-ral 2359  df-v 2540  df-un 2832  df-in 2834  df-sn 3242  df-pr 3243  df-int 3401

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