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Theorem elpr2 3752
 Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)
Hypotheses
Ref Expression
elpr2.1 B V
elpr2.2 C V
Assertion
Ref Expression
elpr2 (A {B, C} ↔ (A = B A = C))

Proof of Theorem elpr2
StepHypRef Expression
1 elprg 3750 . . 3 (A {B, C} → (A {B, C} ↔ (A = B A = C)))
21ibi 232 . 2 (A {B, C} → (A = B A = C))
3 elpr2.1 . . . . . 6 B V
4 eleq1 2413 . . . . . 6 (A = B → (A V ↔ B V))
53, 4mpbiri 224 . . . . 5 (A = BA V)
6 elpr2.2 . . . . . 6 C V
7 eleq1 2413 . . . . . 6 (A = C → (A V ↔ C V))
86, 7mpbiri 224 . . . . 5 (A = CA V)
95, 8jaoi 368 . . . 4 ((A = B A = C) → A V)
10 elprg 3750 . . . 4 (A V → (A {B, C} ↔ (A = B A = C)))
119, 10syl 15 . . 3 ((A = B A = C) → (A {B, C} ↔ (A = B A = C)))
1211ibir 233 . 2 ((A = B A = C) → A {B, C})
132, 12impbii 180 1 (A {B, C} ↔ (A = B A = C))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {cpr 3738 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-sn 3741  df-pr 3742 This theorem is referenced by:  elopk  4129
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