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Theorem uniprg 3906
 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 3799 . . . 4 (x = A → {x, y} = {A, y})
21unieqd 3902 . . 3 (x = A{x, y} = {A, y})
3 uneq1 3411 . . 3 (x = A → (xy) = (Ay))
42, 3eqeq12d 2367 . 2 (x = A → ({x, y} = (xy) ↔ {A, y} = (Ay)))
5 preq2 3800 . . . 4 (y = B → {A, y} = {A, B})
65unieqd 3902 . . 3 (y = B{A, y} = {A, B})
7 uneq2 3412 . . 3 (y = B → (Ay) = (AB))
86, 7eqeq12d 2367 . 2 (y = B → ({A, y} = (Ay) ↔ {A, B} = (AB)))
9 vex 2862 . . 3 x V
10 vex 2862 . . 3 y V
119, 10unipr 3905 . 2 {x, y} = (xy)
124, 8, 11vtocl2g 2918 1 ((A V B W) → {A, B} = (AB))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∪ 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-rex 2620  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: (None)
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