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Theorem cnvun 5033
 Description: The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by set.mm contributors, 27-Aug-2011.)
Assertion
Ref Expression
cnvun (AB) = (AB)

Proof of Theorem cnvun
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unopab 4638 . . 3 ({x, y yAx} ∪ {x, y yBx}) = {x, y (yAx yBx)}
2 brun 4692 . . . 4 (y(AB)x ↔ (yAx yBx))
32opabbii 4626 . . 3 {x, y y(AB)x} = {x, y (yAx yBx)}
41, 3eqtr4i 2376 . 2 ({x, y yAx} ∪ {x, y yBx}) = {x, y y(AB)x}
5 df-cnv 4785 . . 3 A = {x, y yAx}
6 df-cnv 4785 . . 3 B = {x, y yBx}
75, 6uneq12i 3416 . 2 (AB) = ({x, y yAx} ∪ {x, y yBx})
8 df-cnv 4785 . 2 (AB) = {x, y y(AB)x}
94, 7, 83eqtr4ri 2384 1 (AB) = (AB)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   = wceq 1642   ∪ cun 3207  {copab 4622   class class class wbr 4639  ◡ccnv 4771 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-opab 4623  df-br 4640  df-cnv 4785 This theorem is referenced by:  rnun  5036  f1oun  5304
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