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Theorem uniprg 3569
 Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)
Assertion
Ref Expression
uniprg ((A 𝑉 B 𝑊) → {A, B} = (AB))

Proof of Theorem uniprg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 3421 . . . 4 (x = A → {x, y} = {A, y})
21unieqd 3565 . . 3 (x = A {x, y} = {A, y})
3 uneq1 3067 . . 3 (x = A → (xy) = (Ay))
42, 3eqeq12d 2036 . 2 (x = A → ( {x, y} = (xy) ↔ {A, y} = (Ay)))
5 preq2 3422 . . . 4 (y = B → {A, y} = {A, B})
65unieqd 3565 . . 3 (y = B {A, y} = {A, B})
7 uneq2 3068 . . 3 (y = B → (Ay) = (AB))
86, 7eqeq12d 2036 . 2 (y = B → ( {A, y} = (Ay) ↔ {A, B} = (AB)))
9 vex 2538 . . 3 x V
10 vex 2538 . . 3 y V
119, 10unipr 3568 . 2 {x, y} = (xy)
124, 8, 11vtocl2g 2594 1 ((A 𝑉 B 𝑊) → {A, B} = (AB))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374   ∪ cun 2892  {cpr 3351  ∪ cuni 3554 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555 This theorem is referenced by:  onun2  4166
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