Theorem unipr 2511

Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16.

Hypotheses

Ref	Expression
unipr.1	|- A e. V
unipr.2	|- B e. V

Assertion

Ref	Expression
unipr	|- U.{A, B} = (A u. B)

Proof of Theorem unipr

Step	Hyp	Ref	Expression
1		19.43 1087	. . . 4 |- (E.y((x e. y /\ y = A) \/ (x e. y /\ y = B)) <-> (E.y(x e. y /\ y = A) \/ E.y(x e. y /\ y = B)))
2		visset 1810	. . . . . . . 8 |- y e. V
3	2	elpr 2421	. . . . . . 7 |- (y e. {A, B} <-> (y = A \/ y = B))
4	3	anbi2i 480	. . . . . 6 |- ((x e. y /\ y e. {A, B}) <-> (x e. y /\ (y = A \/ y = B)))
5		andi 603	. . . . . 6 |- ((x e. y /\ (y = A \/ y = B)) <-> ((x e. y /\ y = A) \/ (x e. y /\ y = B)))
6	4, 5	bitr 173	. . . . 5 |- ((x e. y /\ y e. {A, B}) <-> ((x e. y /\ y = A) \/ (x e. y /\ y = B)))
7	6	exbii 1050	. . . 4 |- (E.y(x e. y /\ y e. {A, B}) <-> E.y((x e. y /\ y = A) \/ (x e. y /\ y = B)))
8		unipr.1	. . . . . . 7 |- A e. V
9	8	clel3 1890	. . . . . 6 |- (x e. A <-> E.y(y = A /\ x e. y))
10		exancom 1053	. . . . . 6 |- (E.y(y = A /\ x e. y) <-> E.y(x e. y /\ y = A))
11	9, 10	bitr 173	. . . . 5 |- (x e. A <-> E.y(x e. y /\ y = A))
12		unipr.2	. . . . . . 7 |- B e. V
13	12	clel3 1890	. . . . . 6 |- (x e. B <-> E.y(y = B /\ x e. y))
14		exancom 1053	. . . . . 6 |- (E.y(y = B /\ x e. y) <-> E.y(x e. y /\ y = B))
15	13, 14	bitr 173	. . . . 5 |- (x e. B <-> E.y(x e. y /\ y = B))
16	11, 15	orbi12i 257	. . . 4 |- ((x e. A \/ x e. B) <-> (E.y(x e. y /\ y = A) \/ E.y(x e. y /\ y = B)))
17	1, 7, 16	3bitr4r 184	. . 3 |- ((x e. A \/ x e. B) <-> E.y(x e. y /\ y e. {A, B}))
18	17	abbii 1573	. 2 |- {x | (x e. A \/ x e. B)} = {x | E.y(x e. y /\ y e. {A, B})}
19		df-un 2047	. 2 |- (A u. B) = {x | (x e. A \/ x e. B)}
20		df-uni 2500	. 2 |- U.{A, B} = {x | E.y(x e. y /\ y e. {A, B})}
21	18, 19, 20	3eqtr4r 1504	1 |- U.{A, B} = (A u. B)

Syntax hints:   \/ wo 222   /\ wa 223   = wceq 955   e. wcel 957  E.wex 979  {cab 1462  Vcvv 1808   u. cun 2042  {cpr 2407  U.cuni 2499

This theorem is referenced by:  uniprg 2512  unisn 2513  uniop 2804  unex 2868  rankxplim 4695  indistop 7608  indistps 7613  dfchj3 9280  mapudiscn 10458

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-v 1809  df-un 2047  df-sn 2409  df-pr 2410  df-uni 2500

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