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Theorem uniprg 3586
 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 3438 . . . 4 (x = A → {x, y} = {A, y})
21unieqd 3582 . . 3 (x = A {x, y} = {A, y})
3 uneq1 3084 . . 3 (x = A → (xy) = (Ay))
42, 3eqeq12d 2051 . 2 (x = A → ( {x, y} = (xy) ↔ {A, y} = (Ay)))
5 preq2 3439 . . . 4 (y = B → {A, y} = {A, B})
65unieqd 3582 . . 3 (y = B {A, y} = {A, B})
7 uneq2 3085 . . 3 (y = B → (Ay) = (AB))
86, 7eqeq12d 2051 . 2 (y = B → ( {A, y} = (Ay) ↔ {A, B} = (AB)))
9 vex 2554 . . 3 x V
10 vex 2554 . . 3 y V
119, 10unipr 3585 . 2 {x, y} = (xy)
124, 8, 11vtocl2g 2611 1 ((A 𝑉 B 𝑊) → {A, B} = (AB))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   ∪ cun 2909  {cpr 3368  ∪ cuni 3571 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572 This theorem is referenced by:  onun2  4182
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