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Theorem unipr 3905
 Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)
Hypotheses
Ref Expression
unipr.1 A V
unipr.2 B V
Assertion
Ref Expression
unipr {A, B} = (AB)

Proof of Theorem unipr
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . . . . 8 y V
21elpr 3751 . . . . . . 7 (y {A, B} ↔ (y = A y = B))
32anbi2i 675 . . . . . 6 ((x y y {A, B}) ↔ (x y (y = A y = B)))
4 andi 837 . . . . . 6 ((x y (y = A y = B)) ↔ ((x y y = A) (x y y = B)))
53, 4bitri 240 . . . . 5 ((x y y {A, B}) ↔ ((x y y = A) (x y y = B)))
65exbii 1582 . . . 4 (y(x y y {A, B}) ↔ y((x y y = A) (x y y = B)))
7 19.43 1605 . . . 4 (y((x y y = A) (x y y = B)) ↔ (y(x y y = A) y(x y y = B)))
86, 7bitri 240 . . 3 (y(x y y {A, B}) ↔ (y(x y y = A) y(x y y = B)))
9 eluni 3894 . . 3 (x {A, B} ↔ y(x y y {A, B}))
10 elun 3220 . . . 4 (x (AB) ↔ (x A x B))
11 unipr.1 . . . . . . 7 A V
1211clel3 2977 . . . . . 6 (x Ay(y = A x y))
13 exancom 1586 . . . . . 6 (y(y = A x y) ↔ y(x y y = A))
1412, 13bitri 240 . . . . 5 (x Ay(x y y = A))
15 unipr.2 . . . . . . 7 B V
1615clel3 2977 . . . . . 6 (x By(y = B x y))
17 exancom 1586 . . . . . 6 (y(y = B x y) ↔ y(x y y = B))
1816, 17bitri 240 . . . . 5 (x By(x y y = B))
1914, 18orbi12i 507 . . . 4 ((x A x B) ↔ (y(x y y = A) y(x y y = B)))
2010, 19bitri 240 . . 3 (x (AB) ↔ (y(x y y = A) y(x y y = B)))
218, 9, 203bitr4i 268 . 2 (x {A, B} ↔ x (AB))
2221eqriv 2350 1 {A, B} = (AB)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207  {cpr 3738  ∪cuni 3891 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892 This theorem is referenced by:  uniprg  3906  unisn  3907  uniintsn  3963
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