Theorem intunsn 2569

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 2566	. 2 |- |^|(A u. {B}) = (|^|A i^i |^|{B})
2		intunsn.1	. . . 4 |- B e. V
3	2	intsn 2568	. . 3 |- |^|{B} = B
4	3	ineq2i 2217	. 2 |- (|^|A i^i |^|{B}) = (|^|A i^i B)
5	1, 4	eqtr 1498	1 |- |^|(A u. {B}) = (|^|A i^i B)

Syntax hints:   = wceq 958   e. wcel 960  Vcvv 1814   u. cun 2048   i^i cin 2049  {csn 2413  |^|cint 2537

This theorem is referenced by:  fiint 4572  fiintOLD 4573

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-un 2053  df-in 2054  df-sn 2416  df-pr 2417  df-int 2538

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