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Theorem elprg 3750
 Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
Assertion
Ref Expression
elprg (A V → (A {B, C} ↔ (A = B A = C)))

Proof of Theorem elprg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2359 . . 3 (x = A → (x = BA = B))
2 eqeq1 2359 . . 3 (x = A → (x = CA = C))
31, 2orbi12d 690 . 2 (x = A → ((x = B x = C) ↔ (A = B A = C)))
4 dfpr2 3749 . 2 {B, C} = {x (x = B x = C)}
53, 4elab2g 2987 1 (A V → (A {B, C} ↔ (A = B A = C)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {cpr 3738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 This theorem is referenced by:  elpr  3751  elpr2  3752  elpri  3753  eltpg  3769  ifpr  3774  prid1g  3825
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